Nodal count on a family of quantum graphs

Dynamics of nodal points and the nodal count on a family of quantum graphs

Abstract.

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the -th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph’s eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

1. Introduction

Spectral properties of differential operators on graphs have recently arisen as models for such diverse areas of research as quantum chaos, photonic crystals, quantum wires and nanostructures. We refer the interested reader to the reviews [1, 2] as well as to collections of recent results [3, 4]. As a part of this research program, the study of eigenfunctions, and in particular, their nodal domains is an exciting and rapidly developing research direction. It is an extension to graphs of the investigations of nodal domains on manifolds, which started already in the 19th century by the pioneering work of Chladni on the nodal structures of vibrating plates. Counting nodal domains started with Sturm’s oscillation theorem which states that a vibrating string is divided into exactly nodal intervals by the zeros of its -th vibrational mode. In an attempt to generalize Sturm’s theorem to manifolds in more than one dimension, Courant formulated his nodal domains theorem for vibrating membranes, which bounds the number of nodal domains of the -th eigenfunction by [5]. Pleijel has shown later that Courant’s bound can be realized only for finitely many eigenfunctions [6]. The study of nodal domains counts was revived after Blum et al have shown that nodal count statistics can be used as a criterion for quantum chaos [7]. A subsequent paper by Bogomolny and Schmit illuminated the fascinating connection between nodal statistics and percolation theory [8]. A recent paper by Nazarov and Sodin addresses the counting of nodal domains of eigenfunctions of the Laplacian on [9]. They prove that on average, the number of nodal domains increases linearly with , and the variance about the mean is bounded. At the same time, it was shown that the nodal sequence - the sequence of numbers of nodal domains ordered by the corresponding spectral parameters - stores geometrical information about the domain [10]. Moreover, there is a growing body of numerical and theoretical evidence which shows that the nodal sequence can be used to distinguish between isospectral manifolds [11, 12, 13].

As far as counting nodal domains on graphs is concerned, it has been shown that trees behave as one-dimensional manifolds, and the analogue of Sturm’s oscillation theory applies [14, 15, 16, 17], as long as the eigenfunction does not vanish at any vertex. Thus, denoting by the number of nodal domains of the ’th eigenfunction, one has for tree graphs. Courant’s theorem applies for the eigenfunctions on a generic graph: , [18]. It should be noted that there is a correction due to multiplicity of the -th eigenvalue and the upper bound becomes , where is the multiplicity [19]. In addition, a lower bound for the number of nodal domains was discovered recently. It is shown in [20] that the nodal domains count of the -th eigenfunction has no less than nodal domains, where is the number of independent cycles in the graph. Again, this result is valid for generic eigenfunctions, namely, the eigenfunction has no zero entries on the vertices and belongs to a simple eigenvalue. In a few cases, the nodal counts of isospectral quantum graphs were shown to be different, and thus provided further support to the conjecture that nodal count resolves isospectrality [21]. A recent review entitled “Nodal domains on graphs - How to count them and why?” [22] provides a detailed answer to the question which appears in its title (as it was known when the article was written). In particular, this manuscript contains a numerically established formula for the nodal count of a specific quantum graph, expressed in terms of the lengths of its edges. This was the first, and to this date the only, explicit nodal count formula for a non-trivial graph and in this manuscript we succeed in rigorously proving it.

This leads us to focus on the study of nodal domains on quantum graphs from a new point of view. Namely, we shall show that one can count the number of nodal domains by using scattering data obtained by attaching semi-infinite leads to the graph. Scattering on graphs was proposed as a paradigm for chaotic scattering in [23, 24] with new applications and further developments in the field described in [25, 26, 27]. The work presented here is based on the concepts and ideas developed in these studies.

The paper is organized in the following way. The current section provides the necessary definitions and background from the theory of quantum graphs. The conversion of finite graphs to scattering systems by adding leads will be discussed in the next section and the expression for the scattering matrix will be derived and studied in detail. The connection of the scattering data with nodal domains and the counting methods it yields will be presented in section 3. Section 4 applies the above counting methods in order to derive a formula for the nodal count of graphs with disjoint cycles. This formula relates the nodal count to the spectra of the graph and some of its subgraphs. Thus, information about the eigenfunctions is exclusively obtained from the eigenvalue spectrum. The last section relates the different ways of counting and discusses possible future developments.

1.1. Quantum graphs

In this section we describe the quantum graph which is a metric graph with a Shrödinger-type self-adjoint operator defined on it. Let be a connected graph with vertices and edges . The sets and are required to be finite.

We are interested in metric graphs, i.e. the edges of are -dimensional segments with finite positive lengths . On the edge we use two coordinates, and . The coordinate measures the distance along the edge starting from the vertex ; is defined similarly. The two coordinates are connected by . Sometimes, when the precise nature of the coordinate is unimportant, we will simply write or even .

A metric graph becomes quantum after being equipped with an additional structure: assignment of a self-adjoint differential operator. This operator will be often called the Hamiltonian. In this paper we study the zeros of the eigenfunctions of the negative second derivative operator ( is the coordinate along an edge)

(1.1)

or the more general Schrödinger operator

(1.2)

where is a potential. Note that the value of a function or the second derivative of a function at a point on the edge is well-defined, thus it is not important which coordinate, or is used. This is in contrast to the first derivative which changes sign according to the direction of the chosen coordinate.

To complete the definition of the operator we need to specify its domain.

Definition 1.1.

We denote by the space

which consists of the functions on that on each edge belong to the Sobolev space . The restriction of to the edge is denoted by . The norm in the space is

Note that in the definition of the smoothness is enforced along edges only, without any junction conditions at the vertices at all. However, the standard Sobolev trace theorem (e.g., [28]) implies that each function and its first derivative have well-defined values at the endpoints of the edge . Since the direction is important for the first derivative, we will henceforth adopt the convention that, at an end-vertex of an edge , the derivative is calculate into the edge and away from the vertex. That is the coordinate is chosen so that the vertex corresponds to .

To complete the definition of the operator we need to specify its domain. All conditions that lead to the operator (1.1) being self-adjoint have been classified in [29, 30, 31]. We will only be interested in the so-called extended -type conditions, since they are the only conditions that guarantee continuity of the eigenfunctions, something that is essential if one wants to study changes of sign of the said eigenfunctions.

Definition 1.2.

The domain of the operator (1.2) consists of the functions such that

  1. is continuous on every vertex:

    (1.3)

    for every vertex and edges and that have as an endpoint.

  2. the derivatives of at each vertex satisfy

    (1.4)

    where is the set of edges incident to .

Sometimes the condition (1.4) is written in a more robust form

(1.5)

which is also meaningful for infinite values of . Henceforth we will understand as the Dirichlet condition . The case is often referred to as the Neumann-Kirchhoff condition.

Finally, we will assume that the potential is bounded and piecewise continuous. To summarize our discussion, the operator (1.2) with the domain is self-adjoint for any choice of real . Since we only consider compact graphs, the spectrum is real, discrete and with no accumulation points. We will slightly abuse notation and denote by the spectrum of an operator defined on the graph . It will be clear from the context which operator we mean and what are the vertex conditions.

The eigenvalues satisfy the equation

(1.6)

It can be shown that under the conditions specified above the operator is bounded from below [31]. Thus we can number the eigenvalues in the ascending order, starting with . Sometimes we abuse the notation further and also call , such that , an eigenvalue of the graph . This also should lead to no confusion since, with the conditions , , the relation between and is bijective.

1.2. Nodal count

The main purpose of this article is to investigate the number of zeros and the number of nodal domains of the eigenfunctions of a connected quantum graph. We aim to give formulas linking these quantities to the geometry of the graphs and to the eigenvalues of the graph and its subgraphs, but avoiding any reference to the values of the eigenfunctions themselves.

The number of internal zeros or nodal points of the function will be denoted by . We will use the shorthand to denote where is the -th eigenfunction of the graph in question. The sequence will be called the nodal point count sequence. A positive (negative) domain with respect to is a maximal connected subset in where is positive (correspondingly, negative). The total number of positive and negative domains will be called the nodal domain count of and denoted by . Similarly to , we use as a short-hand for and refer to as the nodal domain count sequence.

The two quantities and are closely related, although, due to the graph topology, the relationship is more complex than on a line, where . Namely, one can easily establish the bound

(1.7)

where is the cyclomatic number of . The cyclomatic number can be computed as

(1.8)

The cyclomatic number has several related interpretations: it counts the number of independent cycles in the graph (hence the name) and therefore it is the first Betti number of (hence the notation ). It also counts the minimal number of edges that need to be removed from to turn it into a tree. Correspondingly, if and only if is a tree.

There is another simple but useful observation relating the cycles on the graph and the number of zeros: if the eigenfunction of the graph does not vanish on the vertices of the graph, the number of zeros on any cycle of the graph is even. Indeed, an eigenfunction of a second order operator can only have simple zeros, thus at every zero changes sign. On a cycle there must be an even number of sign changes.

As mentioned earlier, we will be interested in the number of zeros and nodal domains of the eigenfunctions of operators (1.1) and (1.2) on graphs. According to the well known ODE theorem by Sturm [32, 33, 34], the zeros of the -th eigenfunction of the operator of type (1.2) on an interval divide the interval into nodal domains. By contrast, in the corresponding question in only an upper bound is possible, given by the Courant’s nodal line theorem [5], . In a series of papers [14, 15, 18, 17, 20], it was established that a generic eigenfunction of the quantum graph satisfies both an upper and a lower bound. Namely, let be a simple eigenvalue of on a graph and its eigenfunction be non-zero at all vertices of . Then the number of the nodal domains of satisfies

(1.9)

Similarly, for the number of zeros we have

(1.10)

Note that the upper bound in (1.10) follows from the upper bound in (1.9) and inequality (1.7). The lower bound in (1.10) requires an independent proof which is given in [35]. An interesting feature of quantum graphs is that, unlike the case, the upper bound is in general not valid for degenerate eigenvalues.

In the present paper we combine these a priori bounds with scattering properties of a certain family of graphs to derive formulas for the nodal counts and .

1.3. Quantum evolution map

When the potential is equal to zero, the eigenvalue equation

(1.11)

has, on each edge, a solution that is a linear combination of the two exponents if . We will write it in the form

(1.12)

where the variable measures the distance from the vertex of the edge . The coefficient is the incoming amplitude on the edge (with respect to the vertex ) and is correspondingly the outgoing amplitude. However the same function can be expressed using the coordinate as

(1.13)

Since these two expressions should define the same function and since the two coordinates are connected, through the identity , we arrive to the following relations

(1.14)

Fixing a vertex of degree and using (1.12) to describe the solution on the edges adjacent to we obtain from (1.3) and (1.4) equations on the variables and . These equations can be rearranged as

(1.15)

where and are the vectors of the corresponding coefficients and is a unitary matrix. The matrix is called the vertex-scattering matrix, it depends on for values of other than or and its entries have been calculated in [36].

Collect all coefficients into a vector of size and define the matrix acting on by requiring that it swaps around and for all . Then, collecting equations (1.15) into one system and using connection (1.14) and the matrix to rewrite everything in terms of we have

Here all matrices have the dimension equal to double the number of edges, . The matrix is the diagonal matrix of edge lengths, each length appearing twice and is the block-diagonalizable matrix with individual as blocks, namely

Noting that , the condition on can be rewritten as

(1.16)

The unitary matrix is variously called the bond scattering matrix [36] or the quantum evolution map [2]. The matrix describes the scattering of the waves on the vertices of the graph and gives the phase shift acquired by the waves while traveling along the edges. The quantum evolution map can be used to compute the non-zero eigenvalues of the graph through the equation

(1.17)

We stress that is not a scattering matrix in the conventional sense, since the graph is not open. Turning graph into a scattering system is the subject of the next section.

2. Attaching infinite leads to the graph

A quantum graph may be turned into a scattering system by attaching any number of infinite leads to some or all of its vertices. This idea was already discussed in [36, 24, 37]. We repeat it here and further investigate the analytic and spectral properties of the graph’s scattering matrix, that would enable the connection to the nodal count.

Let be a quantum graph. We choose some out of its vertices and attach to each of them an infinite lead. We call these vertices, the marked vertices, and supply them with the same vertex conditions as they had in . Namely, each marked vertex retains its -type condition with the same parameter (recall (1.4)). We denote the extended graph that contains the leads by and investigate its generalized eigenfunctions.

The solution of the eigenvalue equation, (1.11), on a lead which is attached to the vertex , can be written in the form

(2.1)

The variable measures the distance from the vertex along the lead and the coefficients are the incoming and outgoing amplitudes on the lead (compare with (1.12)). We use the notation , for the vectors of the corresponding coefficients and follow the derivation that led to (1.16) in order to obtain

(2.2)

All the matrices above are square matrices of dimension . There are two differences from equation (1.16). First, in the matrix each lead is represented by a single zero on the diagonal, in contrast to the positive lengths of the graph edges, appearing twice each. The matrix swaps around the coefficients corresponding to opposite directions on internal edges, but acts as an identity on the leads. These differences arise because for an infinite lead we do not have two representations (1.12) and (1.13) and therefore no connection formulas (1.14) allowing to eliminate outgoing coefficients. Writing the matrix in blocks corresponding to the edge coefficients and lead coefficients results in

(2.3)

where the dimensions of the matrices , , and are , , and correspondingly. We stress that the matrix describes the evolution of the waves inside the compact graph and has eigenvalues that can now lie inside the unit circle due to the “leaking” of the waves into the leads.

Equation (2.3) can be used to define a unitary scattering matrix such that , as described in the following theorem.

Theorem 2.1.

Let

(2.4)

is a unitary matrix with the blocks , , and of sizes , , and correspondingly. For every choice of , consider relation (2.4) as a set of linear equations in the variables and . Then

  1. There exists at least one matrix such that

    (2.5)
  2. Let

    (2.6)

    Then is a unitary matrix independent of the particular choice of in equation (2.5).

  3. The solutions of (2.3) are given by

    (2.7)
    (2.8)

    In particular, is defined uniquely by .

The proof of the theorem distinguishes between the case of a trivial and the case of singular . The following lemma makes the treatment of the latter case easier.

Lemma 2.2.

Let be as in theorem 2.1. Then the following hold:

(2.9)
(2.10)
Proof.

Since in a finite-dimensional space , equation (2.9) is equivalent to

which is in turn equivalent to

Let . Using the unitarity of we get

(2.11)

Equating the left-hand side to the right-hand side of the equation above we get . Equation (2.10) is proved in a similar manner by replacing with in the above. ∎

Proof of theorem 2.1.

Case 1: .

To show part (1) we simply set . Furthermore, equation (2.3) has a unique solution, given by

(2.12)
(2.13)

which proves part (3).

The unique definition of guarantees the uniqueness of . To finish the proof of part (2) we use the unitarity of , which provides the identities

(2.14)

From here we get

Expanding, factorizing and using the definition of in the form , we arrive to

Case 2: .

Existence of a solution to the equation is guaranteed by equation (2.9) of Lemma 2.2.

The columns of are defined up to addition of arbitrary vectors from . However, Lemma 2.2, equation (2.10) implies that these vectors are in the null-space of , therefore the product has unique value independent of the particular choice of the solution . The proof of the unitarity of has already been given in case 1 and did not rely on the invertibility of . This proves part (2).

The last equations of (2.3) are . From (2.5), all solutions of this equation are given by . On the other hand, the first equations of (2.3) are and substituting the already obtained expression for and using (2.10) we finally arrive to . This finished the proof of the theorem. ∎

We would like to study the unitary scattering matrices, as a one-parameter family in . The matrix is a meromorphic function of in the entire complex plane [29]. For all values which satisfy , is given explicitly by

(2.15)

and is therefore also a meromorphic function in at these values. The significance of the values of for which is explained in the following lemma.

Lemma 2.3.

Let be the quantum graph obtained from the original compact quantum graph, , by imposing the condition at all of its marked vertices, in addition to the conditions already imposed there. Then the spectrum coincides with the set

(2.16)
Proof.

We mention that imposition of the additional vertex conditions makes the problem overdetermined. In most circumstances the set will be empty. The operator is still symmetric but no longer self-adjoint, because its domain is too narrow.

Denote by the graph with the leads attached. Let and let be the corresponding eigenfunction on . Then can be extended to the leads by zero. It will still satisfy the vertex conditions of and will therefore satisfy (2.3) with and . The last equations of (2.3) imply .

In the other direction, let . Choose . We see that equation (2.3) is satisfied with the chosen and with . These coefficients describe a function on which vanishes completely on the leads and therefore its restriction to satisfies Dirichlet boundary conditions on the marked vertices by continuity. This implies . ∎

Corollary 2.4.

The set is discrete.

Proof.

The corollary is immediate since , which is discrete. ∎

Lemma 2.5.

is a meromorphic function which is analytic on the real line.

Proof.

The blocks of the matrix in equation (2.2) are meromorphic (see [29], Theorem 2.1 and the discussion following it), therefore all the blocks of the matrix are meromorphic on the entire complex plane. Since the set on which the matrix is singular is discrete, equation (2.15) defines a meromorphic function. To show that in fact does not have singularities on the real line, we observe, that, for we have shown that defined by (2.15) is unitary. Therefore remains bounded as we approach the “bad” set and the singularities are removable. Theorem 2.1 gives a prescription for computing the correct value of for . ∎

We now examine the -dependence of the eigenvalues of . To avoid technical difficulties we restrict our attention to the case when only (Neumann) or (Dirichlet) are allowed as coefficients of the -type vertex conditions, equation (1.4). In this case the matrix described in section 1.3 is independent of making calculations easier. The general case can be treated using methods of [38], however we will not need it for applications.

Lemma 2.6.

Let every vertex of the graph have either Neumann or Dirichlet condition imposed on it. Then the eigenvalues of move counterclockwise on the unit circle, as increases.

Proof.

Let be an eigenvalue of with the normalized eigenvector . Differentiating the normalization condition with respect to we get

(2.17)

Now we take the derivative of with respect to to get

We multiply the above equation on the left with and use and equation (2.17) to obtain:

Thus we need to show that is positive definite. Comparing equations (2.2) and (2.3) and using that is -independent, we obtain that

Differentiating the latter two matrices with respect to produces

We can now differentiate equation (2.5) to obtain

(2.18)

where we used (2.5) again in the final step.

For the matrix in question we now obtain

where equations (2.14) have been used in the second step. Using which is a conjugate of (2.5), we obtain

Using (2.18) this simplifies to

Since is diagonal with positive entries we conclude that is positive definite. ∎

We end this section by stating a result known as the inside-outside duality, which relates the spectrum of the compact graph, , to the eigenvalues of its scattering matrix, . This is a well known result, mentioned already in [36]. We bring it here with a small modification, related to the already mentioned set, .

Proposition 2.7.

The spectrum of is .

Proof.

We remind the reader that when a lead is attached to a (marked) vertex, the new vertex conditions are also of -type with the same value of the constant . The conditions at the vertices that are not marked remain unchanged.

Let be such that . Let be the corresponding eigenvector, . Letting we find and according to theorem 2.1. The corresponding generalized eigenfunction satisfies correct vertex conditions at all the non-marked vertices. It is also continuous at the marked vertices and satisfies

(2.19)

where is the set of the finite edges incident to and is the set of the infinite leads attached to it. Referring to (2.1) we notice that implies that the derivative of on the lead is zero. Therefore equation (2.19) reduces to the corresponding equation on the compact graph. Thus the restriction of to the compact graph satisfies vertex conditions at all vertices and is an eigenvalue of . Inclusion has already been shown in lemma 2.3.

Conversely, let be an eigenvalue of the compact graph and let be the corresponding eigenfunction. Then can be continued onto the leads as , where is the value of at the vertex where the lead is attached to the graph. Comparing to (2.1) we see that . Therefore the resulting extended function is characterized by vectors , and such that . If the function was non-zero on at least one of the marked vertices, is a valid eigenvector of with eigenvalue 1. If is zero on all marked vertices, by lemma 2.3. ∎

3. Applications to the nodal domains count

3.1. Application for a single lead case.

We wish to study the nodal count sequence of a certain graph by attaching a single lead to one of its vertices. Let be the corresponding one dimensional scattering matrix. For each real there exists a generalized eigenfunction, , of the Laplacian with eigenvalue on the extended graph, , as proved in theorem 2.1. In addition, up to a multiplicative factor, this function is uniquely determined on the lead, where it equals

The positions of the nodal points of this function on the lead are therefore uniquely defined for every real and given by

(3.1)

We exploit this by treating as a continuous parameter and inspecting the change in the positions of the nodal points as increases. Let be the position of a certain nodal point on the lead at some value , i.e., . The direction of movement of this nodal point is given by

(3.2)

where for the last inequality we need to assume that all the vertex conditions of are either of Dirichlet or of Neumann type in order to use the conclusion of lemma 2.6 , . From (3.2) we learn that all nodal points on the lead move towards the graph, as increases.

The event of a nodal point arriving to the graph from the lead occurs at values for which and will be called an entrance event. We may use (3.1) to characterize these events in terms of the scattering matrix:

(3.3)

After such an event occurs, the nodal point from the lead enters the regime of the graph and may change the total number of the nodal points of within .

Another significant type of events is described by

(3.4)

Proposition 2.7 shows that such an event happens at a spectral point of and at this event, the restriction of on equals the corresponding eigenfunction of . These events form the whole spectrum of if and only if . This is indeed the case if we choose to attach the lead to a position where none of the graph’s eigenfunctions vanish (lemma 2.3). In addition, we will assume in the following discussion that has a simple spectrum. This is needed for the unique definition of the nodal count sequences and is shown in [39] to be the generic case for quantum graphs.

The two types of events described by (3.3) and (3.4) interlace, as we know from lemma 2.6 (compare also with theorem A.1). We may investigate the nodal count of tree graphs by merely considering these two types of events and their interlacing property. We count the number of nodal points within only at the spectral points to obtain the sequence . Between each two spectral points we have an entrance event, during which the number of nodal points within increases by one, as a single nodal point has entered from the lead into the regime of . This interlacing between the increments of the number of nodal points and its sampling gives and .

The above conclusion is indeed true for tree graphs under certain assumptions (see [14, 15, 17]). However, when graphs with cycles are considered, there are other interesting phenomena to take into account:

  1. In the paragraph above it was taken for granted that at an entrance event the nodal points count increases by one. This is indeed so if the nodal point which arrives from the lead enters exactly one of the edges of without interacting with other nodal points which already exist on the graph. However, when the lead is attached to a cycle of the graph, the generic behavior is either a split or a merge. Assume for simplicity that the attachment vertex has degree 3, counting the lead. A split event happens when a nodal point from the lead splits into two nodal points that proceed the two internal edges. This will increase the number of nodal points on correspondingly. In a merge event the entering nodal point merges with another nodal point coming along one of the internal edges. The resulting nodal point proceeds along the other internal edge. The number of nodal points on will not change during such an event. If a lead is attached to a vertex of higher degree the variety of scenarios can be larger.

  2. Another type of events that were not considered are ones in which a nodal point travels on the graph and reaches a vertex which is not connected to the lead. When this vertex belongs to a cycle, the generic behavior would be a split or a merge event and would correspondingly increase or decrease the number of nodal points present inside the graph.

These complications are dealt with in section 4.1, where we use the single lead approach to derive a nodal count formula for graphs which contain a single cycle. In the following section, 3.2, we consider a modification of this method — we attach two leads to a graph and use the corresponding scattering matrix to express nodal count related quantities of the graph. Later, in section 3.3, we show how the two leads approach yields an exact nodal count formula for a specific graph.

3.2. Sign-weighted counting function

The number of nodal points on a certain edge at an eigenvalue is given by

(3.5)

where stands for the largest integer which is smaller than , and is the corresponding eigenfunction [18]. We infer that the relative sign of the eigenfunction at two chosen points is of particular interest when counting nodal domains. While the most natural candidates for the two points are end-points of an edge, the results of this section apply to any two points on a graph. Denoting these points and , we are interested in the sign of the product , where is the -th eigenfunction of the graph. We define the sign-weighted counting function as

(3.6)

Using the scattering matrix formalism allows us to obtain the following elegant formula

Theorem 3.1.

Let be a graph with Neumann or Dirichlet vertex conditions and and be points on the graph such that no eigenfunction turns to zero at or . Denote by the scattering matrix obtained by attaching leads to the points and . Let be the matrix

(3.7)

where is the first Pauli matrix. Then

(3.8)

where and a suitable continuous branch of the argument is chosen. The convergence is pointwise everywhere except at (where is discontinuous).

Proof.

First of all, we observe that the scattering matrix of a graph with only Neumann or Dirichlet conditions is complex symmetric: . This can be verified explicitly by using representation (2.5)-(2.6), together with (2.2) and the fact that the matrix is real and symmetric under the specified conditions.

For a moment, consider that only one lead is connected, to the point . Then the events that change the sign of and , where is the one-lead scattering solution, are the values of such that (A) a zero comes into the vertex from the lead (“Dirichlet events”) or (B) a zero crosses the point . The former are easy to characterize: they interlace with the events when a “Neumann point” comes into the vertex , which happen precisely at , as discussed in section 3.1.

Denote by the value of when a zero crosses the point where the lead is not attached. Now consider the scattering system when both leads are attached, at points and . At the one-lead scattering solution can be continued to the second lead by setting it to vanish on the entire lead. This would create a valid two-lead solution with . By inspecting (2.7) we conclude that the vector is therefore an eigenvector of the two-lead scattering matrix . This happens whenever the matrix is diagonal. We conclude that the events of type (B) happen in a one-lead scattering scenario precisely when the two-lead scattering matrix satisfies .

Introducing the notation,

(3.9)
(3.10)

we can summarize the earlier discussion as follows. The eigenvalues of the graph are given by the zeros of (see proposition 2.7), and the relative sign of the -th eigenfunction, , is equal to the parity of the total number of zeros of and that are strictly less than . Note that the condition that no eigenfunction is zero on or implies that the set in proposition 2.7 is empty and that the zeros of the functions and are distinct.

Applying complex conjugation to we obtain

(3.11)

Similarly, using the explicit formula for the inverse of a matrix together with the unitarity of , we obtain for

(3.12)

These relations allow us to represent and , when recalling that .

We now evaluate

and, therefore,

(3.13)

It is now clear that, when (i.e. when ), the limit of the above ratio is a non-zero real number and therefore its argument is an integer multiple of .

To evaluate this integer we focus on the values of when crosses the line . When the crossing is in the counter-clockwise direction, the integer above increases, and otherwise it decreases. The counter-clockwise versus clockwise direction of the crossing is decided exclusively by the sign of the ratio , which coincides with the parity of the total number of zeros of the two functions. This, in turn, has been shown to coincide with the relative sign of the eigenfunction. ∎

Remark 3.2.

It is interesting to compare the above formula for the sign-weighted counting function with the corresponding formula for the more commonly used spectral counting function,

(3.14)

Under conditions of theorem 3.1 the counting function can be represented as

(3.15)

Combining the two we can obtain a counting function that counts only the eigenvalues whose eigenfunctions have differing signs at and ,

(3.16)