Boundary Control for Graphs with Cycles: Dismantling Graphs

Abstract

The goal of this chapter is two-fold: we first describe the general strategy to solve the inverse problems for graphs with cycles; the second part describes how the classical BC-method may be applied to such graphs.

The goal of this chapter is two-fold: we first describe the general strategy to solve the inverse problems for graphs with cycles; the second part describes how the classical BC-method may be applied to such graphs.

21.1 Inverse Problems for Graphs with Cycles: Boundary Control Versus Magnetic Boundary Control

This and the following two chapters are devoted to the solution of the inverse problem for graphs with cycles. We shall always assume that together with the metric graph $\Gamma$ we fix a certain set of contact vertices $\partial \Gamma$ and use the corresponding M-function as the spectral data. If the contact set is large (for example contains all vertices), then the inverse problem is overdetermined. Therefore it is natural to look for optimal contact sets i.e. the smallest sets ensuring unique solvability of the inverse problem. We already know that for trees the set of all except one boundary vertices is optimal. For example in the case of the 3-star graph, it is enough to know the M-function associated with just two boundary vertices (instead of 4 vertices). It turns out that the optimal sets are difficult to characterise in presence of cycles, therefore we shall often be working with sets which are close to optimal (resembling the set of all boundary vertices in the case of trees). For clarity of the presentation we shall generally assume standard vertex conditions, unless introducing other Hermitian conditions does not influence the result.

In what follows we shall assume that the contact set $\partial \Gamma$ contains all degree one vertices. With this convention the contact set will not be optimal even in the case of trees, to make it optimal only one point has to be removed. On the other hand this convention will drastically simplify our studies for the following reason: using locality of the BC-method all branches in a graph can be reconstructed and the inverse problem may be reduced to a smaller graph. Let $\mathbf T \subset \Gamma$ be any branch (subtree) in the original graph $\Gamma$ with $\vert \partial \mathbf T \vert$ pendant vertices. All except one pendant vertices in $\partial \mathbf T$ also belong to $\partial \Gamma$. Hence the M-function for $\Gamma$, or more precisely its block associated with $\partial \mathbf T \cap \partial \Gamma$, determines the subtree and the potential on it.

Every metric graph with several cycles can be seen as a collection of trees attached to a subgraph without degree one vertices. Such a maximal subgraph will be called the graph’s core (see Fig. 21.1, where all contact vertices are marked by dots distinguishing between the degree one (pendant) vertices (the black dots) and higher degree vertices (the red dots)). The graphs which coincide with their core will be called pendant free as they lack degree one vertices.

Fig. 21.1

A metric graph and its core

As a result the inverse problem is reduced to the graph’s core with the contact set given by all original contact vertices that do not have degree one, and the formerly internal vertices to which the branches were attached. All new contact vertices can be seen as descendants of some degree one contact vertices in the original graph. One may say that the contact set for the core is inherited from the contact set on the original graph. Keeping in mind the reduction just described, we limit our studies to pendant free graphs, unless otherwise explicitly stated.

We have seen that dependence of the spectral properties on the magnetic potential is rather explicit—only fluxes of the magnetic field through the cycles play a role. This is well-known as the celebrated Aharonov-Bohm effect [9, 176]. Therefore the following approach appears attractive: reconstruct the metric graph and electric potential q from the M-function considered as a function not only of the spectral parameter $\lambda$ but also of the magnetic fluxes through the cycles. In this way the contact set required to solve the problem sometimes may be drastically reduced: we shall see examples where one contact vertex is enough to solve the inverse problem for a complicated graph with several cycles. We call this new approach the Magnetic Boundary Control-method (MBC-method) since it uses ideas from the Boundary Control-method (BC-method) enriched by adding nontrivial magnetic fields. As in the case of trees, the connection between the response operator and the M-function will be intensively exploited.

We start our studies in this chapter by solving the inverse problem using the traditional BC-method assuming that magnetic potential is zero. Our driving idea will be reduction of the problem to a tree or a set of trees taking into account that the inverse problem for trees is already solved (see Chap. 20). It is effective to look at the graph globally, assuming that cutting the graph at the contact set turns it into a set of trees. This procedure will be called dismantling graphs and it reduces the inverse problem for general graphs to the inverse problem for trees. To determine the M-functions for subtrees we use two ideas: hierarchy of the M-functions described in Sect. 17.3, and the explicit representations for quantum graph M-functions (17.26) and (17.37). To make this procedure work one has to assume that any two subtrees have at most one vertex in common and that their spectra are disjoint.

After that, in Chaps. 22 and 23, we consider the MBC-method. It seems natural to start with graphs having just one cycle, but it turns out that this case is slightly more difficult than the case of several cycles. Therefore we start by looking at graphs with several cycles in Chap. 22. Our approach is based on an operation we call dissolving vertices. Under this operation a graph $\Gamma$ with a vertex $V^0$ of degree $d_0 \geq 3$ is transformed into a graph $\Gamma '$ with the same set of edges and the same vertices except $V^0$, which is split into $d_0$ degree one vertices. More precisely, the equivalence class corresponding to the vertex $V^0$ is substituted with $d_0$ single-element equivalence classes. All other vertices remain unchanged. In this way all edges joined at $V^0$ become pendant in the new graph. Using the dependence on the magnetic fluxes, the M-function for $\Gamma '$ can be determined, provided $V^0 \in \partial \Gamma$ and that it is not a bottleneck (which we will define in Definition 22.8). Applying the peeling procedure to the pendant edges in $\Gamma '$ the original graph reduces to a strictly smaller subgraph. Repetition of his procedure leads to two possible scenarios:

• the whole graph $\Gamma$ and the potential $q$ on it are reconstructed;

• the process terminates before the whole of $\Gamma$ is reconstructed.

In other words, not all graphs can be reconstructed starting from just one contact vertex—this is not surprising. We call the subgraph which is reconstructable starting from a certain contact vertex by the infiltration domain. To accomplish the reconstruction one has to start again from another contact vertex and repeat the described procedure until the whole of $\Gamma$ is recovered or the remaining subgraph is easy to handle.

Finally in Chap. 23 we solve first the inverse problem for the loop and lasso. This result implies in particular that loops in arbitrary graphs cause non-uniqueness, which cannot be removed using the trick with the magnetic flux. We proceed to arbitrary graphs with one cycle in order to illustrate how the MBC-method works and prove its effectiveness in the case of two vertices on the cycle and its redundancy for any higher number of vertices on the cycle. The reason for redundancy is trivial: any three points on a cycle dismantle it into three parts, each pair having one common vertex, allowing one to solve the inverse problem using the conventional BC-method (without involving magnetic fluxes).

We have thus described our strategy towards solution of the inverse problem for graphs with cycles. To guarantee a unique solution of the problem we shall use two types of additional conditions:

• (1) conditions of a topological nature, satisfied or not satisfied independently of the edge lengths and the value of the potential q;

• (2) generically satisfied conditions having spectral character ensuring that the singularities of partial M-functions do not coincide.

These conditions will not always be optimal, but the necessity of such conditions will be clear from the explicit examples.

21.2 Dismantling Graphs I: Independent Subtrees

21.2.1 General Strategy

In this section we are going to assume that the M-function for a Schrödinger operator on a finite compact pendant free metric graph $\Gamma$ is given. It is associated with a given nonempty set of contact points $\partial \Gamma$. As before our aim is to recover the metric graph, the (electric) potential $q$ and the vertex conditions. We are going to assume standard vertex conditions at all contact vertices, hence only the vertex conditions at the internal vertices (not from $\partial \Gamma$) need to be recovered.

To solve the inverse problem we are going to reduce it to the inverse problem on a collection of (sub)trees spanning the original metric graph. The inverse problem for trees is already solved (see Chap. 20). The reduction can be formally divided into two steps:

• geometric reduction:

  dismantling the original metric graph into a collection of subtrees;

• analytic reduction:

  recovering the M-functions for the subtrees from the M-function for the original graph.

Let us remember that the contact set $\partial \mathbf T$ for a tree $\mathbf T$ contains all its pendant vertices. It is clear that these two steps cannot be separated from each other, especially if one is interested in obtaining optimal results: given a metric graph $\Gamma$ one is interested in finding the smallest contact set $\partial \Gamma$, which guarantees solvability of the inverse problem. Interconnection between the two reductions is sophisticated, hence changing between the two strategies one may obtain rather different results.

The analytic reduction will be based on the explicit formulas (17.26) and (17.37). In order to apply these formulas to inverse problems we shall use the following two fundamental results:

• (1) for each singularity of the M-function at least two diagonal entries are singular, provided the metric graph is a tree;

• (2) the singularities and the corresponding residues uniquely determine the M-functions.

The first of these results follows from the fact that each Dirichlet eigenfunction on a metric tree is non-zero close to at least two pendant vertices.

21.2.2 M-functions and Their Singularities

In this section we prove the aforementioned two important facts concerning M-functions and their singularities.

Lemma 21.1

Let $\mathbf T$ be a metric tree with the Schrödinger operator $L_{q} (\mathbf T)$ determined by standard conditions at the boundary vertices $\partial \mathbf T$ and arbitrary Hermitian conditions at the internal vertices, and let $\mathbf M(\lambda )$ be the M-function associated with all pendant vertices in $\mathbf T$ . Then it holds that

• (1) every Dirichlet eigenfunction on $\mathbf T$ is visible, i.e. the corresponding eigenvalue $\lambda _n^{\mathrm {D}}$ is a singularity of the M-function;

• (2) every Dirichlet eigenfunction $\psi _n^{\mathrm {D}}$ on $\mathbf T$ has non-zero derivatives at (at least) two pendant vertices,

• (3) for every Dirichlet eigenvalue $\lambda _n^{\mathrm {D}}$ at least two diagonal elements of $\mathbf M (\lambda )$ are singular.

Proof

Let us prove the lemma for standard vertex conditions first. Consider any Dirichlet eigenfunction $\psi _n^{\mathrm {D}}$ on the tree $\mathbf T$. Assume that all its normal derivatives vanish at all boundary vertices. This eigenfunction is identically zero on all pendant edges as a solution to the second-order differential equation satisfying zero Cauchy data (both the function value and the derivative are zero at the corresponding pendant vertices). Consider any vertex $V^0$ of degree $d_0$ to which at least $d_0-1$ pendant edges are attached. Standard conditions imply that $\psi _n^{\mathrm {D}} (V^0) = 0$ and the normal derivative on the unique non-pendant edge emanating from $V^0$ is also zero. Hence $\psi _n^{\mathrm {D}}$ is identically equal to zero on that edge as well. Continuing in this way we conclude that $\psi _n^{\mathrm {D}}$ is identically zero on all of $\mathbf T$. Thus statement (1) is proven.

To prove statement (2) it is enough to note that the above procedure can be carried out even in the case where we initially know that $\psi _n^{\mathrm {D}}$ has zero normal derivative at all except one pendant vertices.

Statement (3) then follows from the explicit formula (17.37): if two normal derivatives of $\psi _n^{\mathrm {D}}$ are nonzero, then at least two diagonal elements of $\mathbf M(\lambda )$ have singularities at $\lambda _n^{\mathrm {D}}$.

To prove the lemma for general vertex conditions we remember that we always assume that these conditions at each vertex $V^m$ are given by irreducible unitary matrices $S^m$ via (4.8). Therefore it is enough to show the following fact: let $V^0$ be a vertex of degree $d_0$ and let the eigenfunction $\psi _n^{\mathrm {D}}$ be identically equal to zero on $d_0-1$ edges joined at $V^0$, then the eigenfunction is also identically zero on the remaining edge joined at $V^0$. The function $\psi _n^{\mathrm {D}}$ satisfies the vertex conditions at $V^0$ if either

• the Cauchy data on the remaining edge are zero, or

• one of the vectors from the standard basis in $\mathbb C^{d_0}$ is an eigenvector for $S^m$.

The second possibility does not occur since $S^m$ is always assumed irreducible. Zero Cauchy data implies that the eigenfunction is identically zero on the remaining edge. It remains to repeat the procedure as done above. □

Lemma 21.1 cannot be generalised to include arbitrary graphs with cycles due to possible invisible eigenfunctions with support not overlapping with the contact set.

Assume that all singularities of the M-function are known. Then formula (17.37) allows one to reconstruct the M-function up to a constant matrix $\mathbf M (\lambda ')$. In the following lemma we are going to prove explicit asymptotics for the M-function allowing us to determine the constant matrix $\mathbf M (\lambda ')$.

Lemma 21.2

Under the assumptions of Lemma 21.1 , the following asymptotic representation for the M-function holds:

$$\displaystyle \begin{aligned} {} \mathbf M (-s^2) = - s I + o (1), \quad s \rightarrow \infty. \end{aligned} $$

(21.1)

Proof

Formula (19.12) relates the kernel of the response operator to the M-function. In particular the asymptotics of $\mathbf M(\lambda )$ is determined by the short time behaviour of the response operator. For short times (less than double the length of the shortest pendant edge) the response operator for any tree coincides with the diagonal response operator for a collection of $\vert \partial \mathbf T \vert$ intervals with the potential inherited from the pendant edges. Therefore for short times the response operator can be written as an integral convolution operator with the generalised kernel

$$\displaystyle \begin{aligned} {} - \delta'(t) I + \mathrm{diag}\, \{ r_i (t) \} , \end{aligned} $$

(21.2)

where all $r_i (t)$ are the kernels of the scalar response operators and are locally integrable [43].

To use (19.12) we need to take the Laplace transform. Let us first prove the representation (21.1) in the case that $r_i$ is an $L_1$-function. For any $\epsilon > 0$, by taking $\delta$ sufficiently small, we can ensure that $\int _0^\delta \left | r_i(t) \right | dt \leq \epsilon /2$, leading to

Then for $s \geq - \frac {1}{\delta } \ln \frac {\epsilon }{2 \| r_i \|{ }_{L_1}}$ the second integral is also less than $\epsilon /2$ and $| \hat {r}_i (t) | \leq \epsilon .$ The Laplace transform of $\delta ' (t)$ is just $s .$

Essentially the same calculations lead to formula (21.1) in the case of the interval $[0, \ell ]$ with two contact points $x_1 = \ell$ and $x_2 = 0$:

$$\displaystyle \begin{aligned} {} \mathbf M_{[0, \ell]} (-s^2) = \left( \begin{array}{cc} M_{[0, \ell]}^{11} & M_{[0, \ell]}^{12} \\ M_{[0, \ell]}^{21} & M_{[0, \ell]}^{22} \end{array} \right) = -s I + o(1), \quad s \rightarrow \infty, \end{aligned} $$

(21.3)

where $I$ is the unit $2 \times 2$ matrix. The kernel of the response operator contains $\delta '$-singularities corresponding to the reflections from the opposite endpoints, but these singularities are delayed and therefore do not contribute to the asymptotics of the M-function.

Consider now an arbitrary metric tree $\mathbf T$ with $|\partial \mathbf T |$ pendant vertices. We choose any $\ell > 0$ less than the length of any of the pendant edges. Let us denote by $\mathbf T_2$ the tree obtained from $\mathbf T$ by cutting away intervals of length $\ell$ from each of the pendant edges. Then the original tree $\mathbf T$ can be seen as a union of $\mathbf T_2$ and its complement $\mathbf T_1$ in $\mathbf T$:

$$\displaystyle \begin{aligned} \mathbf T = \mathbf T_1 \cup \mathbf T_2.\end{aligned}$$

The graph $\mathbf T_1$ is just a union of $|\partial \mathbf T |$ edges of length $\ell$ and the corresponding M-function can be written in the block form (in analogy with (21.3)):

$$\displaystyle \begin{aligned} {} \mathbf M_1 (-s^2) = \left( \begin{array}{cc} M_1^{11} & M_1^{12} \\ M_1^{21} & M_1^{22} \end{array} \right) = -s I + o(1), \quad s \rightarrow \infty, \end{aligned} $$

(21.4)

where the matrices $M_1^{ij}$ have dimension $|\partial \mathbf T | \times |\partial \mathbf T |$ and the unit matrix $I$ has dimension $2 |\partial \mathbf T | \times 2 |\partial \mathbf T |$. We are going to use Lemma 18.20 and in particular formula (18.34). In our notations the tree $\mathbf T_1$ has $2 |\partial \mathbf T |$ contact points: the first $|\partial \mathbf T |$ points corresponding to the inner points on the pendant edges and the second $|\partial \mathbf T |$ points forming the contact set for $\mathbf T$. The graph $\mathbf T_2$ has $|\partial \mathbf T |$ contact points, all corresponding to the inner points on the pendant edges in $\mathbf T$. Formula (18.34) does not have block structure in the current case as all contact vertices in $\mathbf T$ come from the contact vertices in $\mathbf T_1$:

$$\displaystyle \begin{aligned} \mathbf M_{\mathbf T} (\lambda) = M_1^{22} (\lambda) - M_1^{21} (\lambda) \left(M_1^{11} (\lambda) + \mathbf M_2(\lambda) \right)^{-1} M_1^{12} (\lambda),\end{aligned}$$

where $\mathbf M_2 (\lambda )$ is the M-function for $\mathbf T_2$. The matrix valued functions $M_1^{11} (-s^2)$ and $M_1^{22} (-s^2)$ are asymptotically close to $-s I$, this follows from (21.4). The explicit representation (17.26) implies that the matrix valued function $\mathbf M_2 (-s^2)$ is negative definite for sufficiently large $s$. Hence the inverse matrix function $\left (M_1^{11} (-s^2) + \mathbf M_2(-s^2) \right )^{-1}$ is uniformly bounded as $s \rightarrow \infty$. It follows that

$$\displaystyle \begin{aligned} M_1^{21} (-s^2) \left(M_1^{11} (-s^2) + \mathbf M_2(-s^2) \right)^{-1} M_1^{12} (-s^2) = o(1), \quad s \rightarrow \infty, \end{aligned}$$

where we have taken into account that $M_1^{21} (-s^2), M_1^{12} (-s^2) = o(1)$, which again follows from (21.4).

The asymptotics of $\mathbf M_{\mathbf T} (-s^2)$ coincides with the asymptotics of $M_1^{22} (-s^2)$ and therefore satisfies (21.1). □

The same result holds for any graph with the contact set given by degree one vertices since our proof was based on the explicit representation for the response operator for short times. We did not use that $\mathbf M_2 (\lambda )$ is associated with a metric tree. To generalise the result to general graphs we need to take into account the degrees of contact vertices.

Lemma 21.3

Let $\Gamma$ be a finite metric graph with the associated Schrödinger operator $L_{q,a} (\Gamma )$ and let $\mathbf M(\lambda )$ be the M-function associated with the set $\partial \Gamma$ of contact vertices. Let $L_{q,a} (\Gamma )$ be defined by standard vertex conditions on $\partial \Gamma$ and arbitrary Hermitian conditions at all other (inner) vertices. Then the following asymptotic representation for the M-function holds:

$$\displaystyle \begin{aligned} {} \mathbf M (-s^2) = - s \; \mathrm{diag} \left\{ d_j \right\} + o (1), \quad s \rightarrow \infty, \end{aligned} $$

(21.5)

where $d_j$ is the degree of the contact vertex $V^j \in \partial \Gamma .$

Proof

The proof of Lemma 21.2 is based on the relation between the asymptotic behaviour of the M-function and short time behaviour of the dynamical response operator. For sufficiently short times the response operator for $\Gamma$ is diagonal and each entry coincides with the response operator for the star graph of degree $d_j$, with the potential inherited from $\Gamma .$ The response operator for any star graph coincides with the sum of the response operators for single edges since we assumed standard vertex conditions at the contact vertices. It follows that the kernel of the response operator for small t possesses the representation

$$\displaystyle \begin{aligned} - \delta'(t) \; \mathrm{diag} \left\{ d_j \right\} + \mathrm{diag} \left\{ r_j (t) \right\} \end{aligned}$$

instead of (21.2). □

The rest of this section is devoted to describing the solution of the inverse problem through the method of dismantling graphs.

21.2.3 Dismantling Graphs I: Independent Subtrees

Our goal in this subsection is to study how to solve the inverse problem by dismantling metric graphs into subtrees assuming that the magnetic potential is zero. Adding a magnetic potential does not help in solving the inverse problem using this method.

Definition 21.4

We say that a set of vertices dismantles a metric graph $\Gamma$ if and only if by completely separating the equivalence classes corresponding to these vertices the graph $\Gamma$ is turned into a collection of (sub)trees $\mathbf T_j$ completely covering $\Gamma$.

Dismantling graphs can be illustrated by introducing Dirichlet conditions at the selected vertices. Assume that the vertex conditions at a vertex $V^0$ of degree $d_0 \neq 1,$ are replaced with Dirichlet conditions. Then the metric graph corresponding to the new operator is not the original one, but the graph with the vertex $V^0$ separated into $d_0$ pendant Dirichlet vertices.

It will be convenient to consider the resulting metric trees $\mathbf T_j$ as subsets of the original graph. Therefore two different trees may have common points (vertices). This happens if their pendant vertices come from the same vertices in the original graph, hence trees with common vertices are unavoidable unless $\Gamma$ is itself a tree. It will be important to distinguish the case where pairs of subtrees have one or several common points.

Definition 21.5

A set of subtrees $\left \{ \mathbf T_j \right \}$ of a metric graph $\Gamma$ is called independent if any pair has at most one common vertex. Otherwise the set is called dependent.

The case of dependent subtrees is more subtle and requires using the Magnetic Boundary Control method—this direction will be investigated in Sect. 23.4. We restrict our studies here to the case of independent subtrees.

Formulating our results we shall separate assumptions having topological nature (enumerated by numbers) from the generically satisfied spectral assumptions (enumerated by letters).

Theorem 21.6

Let $L_{q,0}^{\mathrm {st}} (\Gamma )$ be the standard Schrödinger operator on a pendant free metric graph $\Gamma$ with a selected non-empty contact set $\partial \Gamma$ that dismantles the graph into a set of trees $\{ \mathbf T_j \}$ such that

• (1) no subtree$\mathbf T_j$has two pendant vertices coming from the same vertex in$\Gamma$;

• (2) the subtrees $\mathbf T_j$ are independent, i.e. no two subtrees have more than one common vertex.

Then the M-function associated with the contact set $\partial \Gamma$ generically determines the metric graph $\Gamma$ and the potential $q$ , provided that

• (a) the Schrödinger operators $L_{q,0}^{\mathrm {st}, \mathrm {D} } (\mathbf T_j), \; j =1,2, \dots$ with Dirichlet conditions at the pendant vertices and standard vertex conditions at all internal vertices, have disjoint spectra:

  $$\displaystyle \begin{aligned} \lambda_n^{\mathrm{D}} (\mathbf T_j) \neq \lambda_m^{\mathrm{D}} (\mathbf T_i), \quad j \neq i. \end{aligned} $$

  (21.6)

Proof

The M-function for $\Gamma$ is completely determined by the M-functions for the subtrees $\mathbf T_j.$ In principle the matrix functions $\mathbf M_j (\lambda )$ associated with the subtrees have dimension $|\partial \mathbf T_j| \times | \partial \mathbf T_j |$, but we shall see them as $|\partial \Gamma | \times | \partial \Gamma |$ matrices with zero entries corresponding to contact points from $\partial \Gamma \setminus \partial \mathbf T_j.$ Under this convention the M-function for $\Gamma$ is just equal to the sum of the M-functions for the subtrees:

$$\displaystyle \begin{aligned} {} \mathbf M(\lambda) = \sum_j \mathbf M_j (\lambda). \end{aligned} $$

(21.7)

We denote by $\lambda _n^{\mathrm {D}} (\mathbf T_j)$ the eigenvalues of the Schrödinger operator determined by Dirichlet conditions at pendant vertices on $\mathbf T_j$ and standard conditions at all internal vertices. Formula (17.37) implies that the matrix Herglotz-Nevanlinna functions $\mathbf M_j (\lambda )$ may have singularities at $\lambda _n^{\mathrm {D}} (\mathbf T_j)$, every singularity is present in the case of trees since every eigenfunction is visible (Lemma 21.1, statement (1)). Moreover, at each $\lambda _n^{\mathrm {D}}$ at least two diagonal entries of $\mathbf M_j (\lambda )$ are singular (Lemma 21.1, statement (3)).

All singularities are preserved in $\mathbf M (\lambda )$ since the eigenvalues corresponding to different subtrees are different (assumption a)).

To illustrate the structure of the M-function associated with the original graph, it will be convenient to use colors. Consider the graph presented in Fig. 21.2 with all subtrees coloured (Fig. 21.3).

Fig. 21.2

Dismantling a metric graph. Red dots—contact vertices in $\Gamma$; black dots—contact vertices for subtrees

Fig. 21.3

Coloured subtrees

In the formula for $\mathbf M(\lambda )$ we colour the entries corresponding to different subtrees:

The zero entries are left empty in the formula above.

Our next step is to identify the non-disjoint subsets $\partial \mathbf T_j \subset \partial \Gamma$ and sort the Dirichlet eigenvalues $\lambda _n^{\mathrm {D}} (\Gamma )$ into disjoint subsets corresponding to $\mathbf T_j$. Consider first $\lambda _1^{\mathrm {D}} (\Gamma ).$ This is the lowest Dirichlet eigenvalue and therefore it is a ground state for one of the trees. The corresponding eigenfunction has non-zero derivatives at all pendant vertices of the subtree. Hence taking all contact points from $\partial \Gamma$ for which the corresponding diagonal entry of $\mathbf M (\lambda )$ is singular at $\lambda = \lambda _1^{\mathrm {D}} (\Gamma )$ we get all contact points for one of the trees. Without loss of generality we denote this set by $\partial \mathbf T_1.$

Let us continue with $\lambda _2^{\mathrm {D}} (\Gamma ).$ There are two alternatives:

• (1) $\lambda _2^{\mathrm {D}} (\Gamma )$ is the second lowest Dirichlet eigenvalue on $\mathbf T_1;$

• (2) $\lambda _2^{\mathrm {D}} (\Gamma )$ is the lowest Dirichlet eigenvalue for another subtree.

To distinguish these cases we identify which diagonal entries of $\mathbf M (\lambda )$ are singular at $\lambda = \lambda _2^{\mathrm {D}} (\Gamma ) .$ The first alternative occurs if $\mathbf M(\lambda _2^{\mathrm {D}} (\Gamma ) )$ is singular at the vertices from $\partial \mathbf T_1$ and nowhere else. We get the second alternative if at most one of the singular entries correspond to a vertex from $\partial \mathbf T_1$, and all other entries correspond to vertices outside $\partial \mathbf T_1$. In the second case we form the new set $\partial \mathbf T_2$ containing all vertices where the diagonal elements of $\mathbf M (\lambda )$ are singular at $\lambda = \lambda _2^{\mathrm {D}} (\Gamma )$.

This process can be continued. Assume that the sets $\partial \mathbf T_1, \partial \mathbf T_2, \dots , \partial \mathbf T_m$ have been identified and we are about to consider $\lambda _n^{\mathrm {D}} (\Gamma ) , \; n \geq m$. We again have two alternatives:

• (1) $\lambda _n^{\mathrm {D}}$ is one of the higher Dirichlet eigenvalues on one of the $\mathbf T_j, \, j =1, \dots , m;$

• (2) $\lambda _n^{\mathrm {D}}$ is the lowest Dirichlet eigenvalue for a new subtree $\mathbf T_{m+1}$.

The second alternative is selected if the diagonal of $\mathbf M(\lambda _n^{\mathrm {D}} )$ has at most one singular entry at the vertices corresponding to each $\partial \mathbf T_j, \, j= 1, \dots , m.$ In that case we build a new set $\partial \mathbf T_{m+1}$, otherwise $\lambda _n^{\mathrm {D}}$ is added to the set of Dirichlet eigenvalues for one of the selected subtrees. In a finite number of steps all contact sets $\partial \mathbf T_j$ are identified. Any two sets $\partial \mathbf T_j$ and $\partial \mathbf T_i, \; j \neq i$ have at most one common point. Then all Dirichlet eigenvalues are sorted into the disjoint sets $\{ \lambda _n^{\mathrm {D}} (\mathbf T_j) \}$ using the fact that the eigenfunctions on $\mathbf T_j$ have at least two non-zero normal derivatives at the vertices from $\partial \mathbf T_j$ and the subtrees are independent.

The obtained information together with formula (17.37) (or equivalently (18.7)) allows us to reconstruct $\mathbf M_j (\lambda )$ up to the constant matrices $\mathbf A_j = \mathbf M_j (\lambda ')$:

$$\displaystyle \begin{aligned} {} \mathbf M_j (\lambda) = \mathbf A_j + \sum_{\lambda_n^{\mathrm{D}} (\mathbf T_j)} \frac{\lambda- \lambda'}{(\lambda_n^{\mathrm{D}} - \lambda) (\lambda_n^{\mathrm{D}} - \lambda')} \langle \partial \psi_n^{\mathrm{D}} \vert_{\partial \Gamma}, \cdot \rangle_{\ell_2(\partial \Gamma)} \partial \psi_n^{\mathrm{D}} \vert_{\partial \Gamma}, \end{aligned} $$

(21.8)

subject to

$$\displaystyle \begin{aligned} \sum_j \mathbf A_j = \mathbf M (\lambda'). \end{aligned} $$

(21.9)

It remains to determine the Hermitian matrices $\mathbf A_j.$ The representation (21.1) implies that the matrices $\mathbf A_j$ (as well as the matrix $\mathbf M (\lambda ')$) are uniquely determined provided the singular part in (21.8) is known:

$$\displaystyle \begin{aligned} {} \mathbf A_j = \lim_{s \rightarrow \infty} \left( \sum_{\lambda_n^{\mathrm{D}} (\mathbf T_j)} \frac{s^2 + \lambda'}{(\lambda_n^{\mathrm{D}} +s^2) (\lambda_n^{\mathrm{D}} - \lambda')} \langle \partial \psi_n^{\mathrm{D}} \vert_{\partial \Gamma}, \cdot \rangle_{\ell_2(\partial \Gamma)} \partial \psi_n^{\mathrm{D}} \vert_{\partial \Gamma} - s I_j \right). \end{aligned} $$

(21.10)

Here $I_j$ denotes the diagonal matrix with entries equal to 1 at positions corresponding to vertices in $\partial \mathbf T_j$ and 0 otherwise. Note that the asymptotics of $\mathbf M (\lambda )$ is determined by the degrees $d_m = d (V^m)$ of the contact vertices:

$$\displaystyle \begin{aligned} {} \mathbf M (-s^2) = - \mathrm{diag} \{ d (V^m) \} + o (1), \quad s \rightarrow \infty. \end{aligned} $$

(21.11)

The M-functions for the subtrees determine the subtrees and the potential there (Theorem 20.21). Here it is important that we assumed standard vertex conditions everywhere. Having the subtrees in our hands together with the sets $\partial \mathbf T_j$ allows us to reconstruct the original graph $\Gamma$. The subsets $\partial \mathbf T_j \subset \partial \Gamma$ indicate how to glue together the subtrees. Then the Schrödinger operator on $\Gamma$ is obtained by introducing standard vertex conditions at the contact vertices. □

Let us note that we never explicitly used in the proof that the vertex conditions at the internal vertices are standard, the only fact we needed is that the $\mathbf M_j (\lambda )$ determine the corresponding trees. Theorem 20.21 states that under mild assumptions on the vertex conditions the M-function determines the tree, the potential q and the vertex conditions at internal vertices. Hence we have in fact proved the following stronger result:

Theorem 21.7

Let $L_{q,0}^{\mathbf S} (\Gamma )$ be a Schrödinger operator on a pendant free metric graph $\Gamma$ with a selected non-empty contact set $\partial \Gamma$ that dismantles the graph into a set of trees $\{ \mathbf T_j \}$ such that

• (1) no subtree$\mathbf T_j$has two pendant vertices coming from the same vertex in$\Gamma$;

• (2) the subtrees $\mathbf T_j$ are independent, i.e. any two subtrees have at most one common point.

Let the vertex conditions determining the Schrödinger operator$L_{q,0}^{\mathbf S} (\Gamma )$be generalised delta couplings (see Sect.3.7) at all internal vertices and standard at all contact vertices$\partial \Gamma$.

Assume in addition the following generically satisfied assumption:

• (a) the Schrödinger operators $L_{q,0}^{{\mathbf S}, \mathrm {D} } (\mathbf T_j), \; j =1,2, \dots$ with the vertex conditions at the internal vertices inherited from $\Gamma$ and Dirichlet conditions at the pendant vertices have disjoint spectra

  $$\displaystyle \begin{aligned} \lambda_n^{\mathrm{D}} (\mathbf T_j) \neq \lambda_m^{\mathrm{D}} (\mathbf T_i), \quad j \neq i. \end{aligned} $$

  (21.12)

Then the M-function associated with the contact vertices generically determines the metric graph, the potential $q$ and the vertex conditions at internal vertices.

Problem 92

Check that the proof of Theorem21.6can be adapted to justify Theorem21.7. Point out all necessary changes. It is assumed that the vertex conditions at the inner vertices are generalised delta couplings, why it is not enough to require that conditions at internal vertices are asymptotically properly connecting (see Definition11.4)?

The conditions of the above theorem are not optimal. Let us go through all assumptions of the theorems discussing how they can be weakened without much affecting the proof:

• the standard conditions at the contact vertices may be substituted with any other Hermitian conditions; to be able to recover these conditions one has to assume that any single reflection coefficient is enough for the reconstruction; for example delta couplings may be assumed there;

• the generalised delta couplings at the internal vertices may be substituted with arbitrary asymptotically properly connecting conditions, but the proof has to be modified since we can no longer use that the ground state eigenfunction for the Dirichlet operators on a subtree have non-zero derivatives at all contact vertices (see Problem 93);

• one may allow some of the $\mathbf T_j$’s to have two pendant vertices coming from the same vertex in the original graph—this is equivalent to allowing the subtree to have cycles; remember that the $\mathbf T_j$’s are treated as subsets of the original graph (see Chap. 23);

• one may also allow some pairs of $\mathbf T_j$’s to have more than one common vertex—this case will be studied later on using Magnetic Boundary Control (see Sect. 23.4);

• the requirement that the spectra of $L^{{\mathbf S}, \mathrm {D}}_{q,0} (\mathbf T_j)$ are disjoint can be weakened by allowing common eigenvalues for trees lying far away in $\Gamma$; this is similar to what has been done in [409], reconstructing graphs from their spectra via the trace formula.

As a corollary of Theorem 21.6 we may prove that the graph and the potential on it are uniquely determined by the M-function associated with all vertices.

Theorem 21.8

Let $L_{q,0}^{\mathrm {st}} (\Gamma )$ be the standard Schrödinger operator on a metric graph $\Gamma$ without loops or parallel edges. Assume the generically satisfied condition that

• (a) the Dirichlet Schrödinger operators $L_{q,0}^{\mathrm {D}} (E_n), \, n = 1,2, \dots , N$ on the edges have disjoint spectra

  $$\displaystyle \begin{aligned} \lambda_n^{\mathrm{D}} (E_j) \neq \lambda_m^{\mathrm{D}} (E_i), \quad j \neq i. \end{aligned} $$

  (21.13)

Then the M-function associated with all vertices in$\Gamma$determines the metric graph$\Gamma$and the potential$q$.

Proof

To prove the Theorem we need to check that all conditions of Theorem 21.6 are satisfied. The graph’s edges form the simplest possible trees. Non-existence of loops implies condition (1). The subtrees are independent (condition (2)) because parallel edges are excluded. □

Problem 93

Prove Theorem 21.7 assuming that the conditions at internal vertices are asymptotically properly connecting but are not necessarily generalised delta couplings.