1 Introduction

Spectra of graph neighborhoods and scattering

Abstract.

Let be a family of ’-thin’ Riemannian manifolds modeled on a finite metric graph , for example, the -neighborhood of an embedding of in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on as , for various boundary conditions. We obtain complete asymptotic expansions for the th eigenvalue and the eigenfunctions, uniformly for , in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family .

Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all eigenfunctions are obtained in this way.

Key words and phrases:
Quantum graph, cylindrical end, perturbation theory
2000 Mathematics Subject Classification:
Primary 58J50 35P99 Secondary 47A55 81Q10

1. Introduction

Consider a graph with a finite number of vertices and edges embedded in with straight edges. For let be the set of points of distance at most from . For small , ’looks almost like’ ; in other words, the one-dimensional object should be considered as a good model for , and so one expects many physical and analytical properties of to be understandable, in an approximate sense, by corresponding properties of . The property we analyze in this light here is the spectrum of the Laplacian, under various boundary conditions. For motivation from physics, see for example [13].

This problem has received some attention in the last decade. The case of Neumann boundary conditions was analyzed at various levels of generality in [4], [6], [15], [21], [5], where it was shown that for each the th eigenvalue of the Neumann Laplacian on converges, as , to the th eigenvalue of the second derivative operator on the union of the edges of , where at the vertices so-called Kirchhoff boundary conditions are imposed. The question of what the corresponding limiting behavior is for other, for example Dirichlet, boundary conditions was characterized as ’very difficult’ in [5] and remained open until some partial progress was made recently, see Section 1.4.

In this paper we solve this problem for a general mixed boundary problem, where we impose Dirichlet boundary conditions on one part of the boundary and Neumann conditions on the rest. Other boundary conditions, for example Robin, are also possible. Instead of the setting of an embedded graph described above, we consider the more general, and mathematically more natural, situation of a shrinking family of Riemannian manifolds modeled on the graph: We now consider as an abstract metric graph, that is, for each edge a positive number (to be thought of as half the edge length) is given. In addition, we have geometric data: For each edge an -dimensional Riemannian manifold , and for each edge an -dimensional Riemannian manifold ; all these manifolds are compact and may have a piecewise smooth boundary; also, for each edge incident to a vertex (denoted ), we are given an identification (gluing map) of with a subset of the boundary of , without overlaps. Then is defined by gluing cylinders of length and cross section (the factor denotes a rescaling of the metric) to for all pairs . Also given are subsets D and N of the boundaries of each and , which yield corresponding subsets of (or, for Robin conditions, similar data). We investigate the Laplace-Beltrami operator on , where Dirichlet boundary conditions are imposed on D and Neumann conditions on N. We suppose sufficient regularity of the boundary and the D/N decomposition so that this problem has discrete spectrum.

Besides the -neighborhoods mentioned before this manifold setting includes the case of the boundary of the -neighborhood of (in this case, itself has no boundary). Our results imply the following theorem. {theorem} Let be the eigenvalues of the Laplacian on , with the given boundary conditions, repeated according to their multiplicity.

For each edge , let be the smallest eigenvalue of on , with the given boundary conditions, and let .

There are numbers , and so that for

(1)
(2)

In fact, we obtain much more precise information: complete asymptotics up to errors of the form also in the case , uniform estimates not only for fixed but also for on the order of as well as detailed information on the eigenfunctions, see the theorems below.

For Theorem 1 to be meaningful it is essential to identify the numbers and in terms of the given data. We do this in two steps: First, we give general formulas identifying these numbers in terms of scattering data of an associated non-compact -dimensional limiting problem. Second, we analyze how this scattering data can be obtained from the given data. It turns out that in this second step the situation of general boundary conditions carries some essential new features compared to Neumann boundary conditions: while in this case one has and the are determined by the graph and the edge lengths alone (that is, independent of the manifolds , ), in general and the and depend also on transcendental analytic information involving the manifolds and : The are -eigenvalues in the afore-mentioned limit problem, and the are eigenvalues of a quantum graph whose vertex boundary conditions involve the scattering matrix of that limit problem, see Theorems 1 and 1.1.

1.1. Main results

We now state our results more precisely. Because of the divergence in (1), (2) it is more convenient, and for a proper understanding it turns out to be essential, to rescale the problem: We multiply all lengths by . We denote

This rescales the eigenvalues by the factor . In , the vertex manifolds and the cross sections of the edges are independent of while the lengths of the edges are , and we are interested in the limit . Central to the analysis is the limit object, to be thought of as ,

(3)

where the ’star’ of a vertex of is obtained by attaching a half infinite cylinder to the vertex manifold , for each edge incident to , see Figure 1. Denote by the cross-section for , i.e. the disjoint union of the , each one appearing twice (once for each endpoint). The graph structure is encoded by a map which toggles the two copies of for each edge . See Section 2 for precise definitions.

Figure 1. The setup: and

Since is a non-compact space, the spectrum of the Laplacian (with corresponding boundary conditions) is no longer discrete. Its spectral theory is obtained via scattering theory, that is, one considers it as compact perturbation of , the disjoint union of the edge cylinders. Let be the smallest and second smallest eigenvalue of the Laplacian of and let be the eigenspace corresponding to . The absolutely continuous spectrum of is . To each is associated the scattering matrix , a linear map depending holomorphically on , and to each a scattering solution (generalized eigenfunction) . These are the main objects of scattering theory. In addition, may have discrete spectrum (-eigenvalues).

We also have linear maps where encodes the edge lengths (it equals on the subspace of corresponding to the edge ) and is induced by the graph structure map above. The maps and are involutions on , so they have only the eigenvalues .

We need the following data derived from the spectral data of :

  • The -eigenvalues of , with eigenspaces ;

  • the space .

  • the spaces for . This is non-zero for a discrete set

    (4)
  • for each , pairwise different holomorphic functions , with , defined on and with

    (5)

    for each , and holomorphically varying subspaces with .

The and arise from a bifurcation analysis of , see Theorem 5.2. {theorem}[Main Theorem] Denote by the Laplace-Beltrami operator on , with the given boundary conditions. For any there are such that for the spectrum of in consists of (counting multiplicities) {enuma} many eigenvalues of the form , for , many eigenvalues of the form , many eigenvalues of the form (6) for each and for which . The constants and the constants in the big only depend on the spectral data of . In particular, the asymptotics are uniform for eigenvalues . In terms of (6) gives an eigenvalue by (5), so this implies Theorem 1: take with ; the are the repeated times; and the are repeated times, and the numbers repeated many times. The with are invisible in Theorem 1 since there is fixed as . Theorem 1 also describes eigenvalue with as . We determine the permitted range of . From one obtains , and the Weyl asymptotics for the , see Proposition 5.2, show that this is equivalent to (7) See Theorem 2 for a more explicit description of eigenvalues with close to some . We can also describe the eigenfunctions in terms of the scattering solutions and of the eigenfunctions in , with exponentially small errors, see Theorems 7.1, 7.2 and 7.3 and Corollary 86 for precise statements. The restriction is made to keep the exposition at a reasonable length. Our methods are such that a generalization to higher eigenvalues, taking into account several thresholds, should be fairly straightforward. To identify the numbers in Theorem 1 in terms of the data, we recall that a quantum graph is given by a metric graph ( with the edge lengths ) together with a self-adjoint realization of the operator acting on functions defined on the disjoint union of the edges, where is a variable along each edge, measuring length; such a self-adjoint extension is given by boundary conditions at the vertices. We state the boundary conditions obtained in our setting. Denote by the eigenspace for of the Laplacian on the cross section for an edge . For each vertex let , then clearly corresponding to the decomposition (3). Also, the scattering matrix is the direct sum of scattering matrices . For each vertex , is an involution on . Denote by the restriction of a function on to the edge , and by its inward normal derivative at the endpoint of . {theorem} The numbers in (2) are the eigenvalues of the operator on the metric graph with edge lengths , defined on the space of functions which on each edge are smooth and take values in and which at each vertex satisfy the boundary conditions (8) (9) {remarks} If, for some edge , the lowest eigenvalue on is bigger than , then , so has to be identically zero, which means that the edge may be omitted from . In other words, only the ’thickest’ edges contribute substantially to the eigenvalue , for small . If is connected then is one-dimensional, so it can be identified with . But in general there is no canonical eigenfunction, so this identification is not canonical. However, if all are the same connected Riemannian manifold (for example, in the case of an -neighborhood of an embedded graph, where they are balls) one may choose the same basis element in each , so one may think of as a complex valued function and where is the degree of . Two special cases deserve to be mentioned: Dirichlet conditions at the vertex correspond to . In particular, if then the limit quantum graph is completely decoupled (Dirichlet boundary value problem on each edge separately, without interaction between edges). If all are identified with and has -eigenspace spanned by the vector then (8) says that is continuous at and (9) (which is equivalent to ) that the sum of normal derivatives of the vanishes at . This is often called the ’Kirchhoff boundary condition’. Theorems 1 and 1.1 show clearly the two ingredients that determine the leading behavior of eigenvalues: The determination of and of the -eigenvalues is a transcendental problem, depending on the vertex and edge manifolds (for example, in the situation of the -neighborhood of an embedding, the angle at which edges meet). Given , the are determined solely by the combinatorics of the underlying metric graph. In the special case of pure Neumann boundary conditions and of Dirichlet conditions with ’small’ vertex manifolds the leading behavior of eigenvalues is determined by the metric graph alone, see Theorem 8. That is, it is independent of the manifolds . This may be seen as reason why the general case is harder to analyze. These cases were known previously, see [15], [21], [5] and [20]. It is known that ’usually’ (for example in the embedded situation) there are -eigenvalues , so they can not be neglected. See, for example, [22], [2]. Using Theorem 1.1 one can show that for generic geometric data (for example, an open dense set of Riemannian metrics on the vertex and edge manifolds) one has , that is decoupled Dirichlet conditions for the quantum graph at all vertices. This will be pursued in a separate paper.

1.2. Outline of the proof of the Main Theorem

We now give an outline of the proof of Theorem 1, where for simplicity we assume that all : Let be a coordinate on the cylindrical part of and of , measuring length along the cylinder axis (going from to from both ends of the cylinder axis).

The proof consists of two steps: First, we use the spectral data on to construct approximate eigenfunctions on for large and conclude the existence of eigenvalues and eigenfunctions as claimed. Second, we show that all eigenfunctions on are obtained this way.

For clarity we assume in this outline that all multiplicities (i.e. dimensions of and ) are equal to one and that for all (no embedded eigenvalues). We will call a quantity ’very small’ if it is exponentially small as , i.e. for some .

Throughout, eigenvalues on will be denoted , with if , and spectral values for by , with if .

Case I: Eigenvalues : Here, everything is quite straightforward.

  • First step: If is an eigenfunction on with eigenvalue then decreases exponentially in . So if we simply cut off smoothly near , making it zero near , then we get a function on which satisfies the eigenfunction equation up to a very small error. A simple spectral approximation lemma, Lemma 37, shows that has an eigenvalue very close to . A spectral gap argument gives a similar approximation for the eigenfunction.

  • Second step: This procedure can be reversed: If is an eigenfunction on with eigenvalue then it decreases exponentially in , so cutting it off near yields a function on which satisfies the eigenfunction equation up to a very small error. Since has purely discrete spectrum near , it follows by the spectral approximation lemma that it has an eigenvalue very close to . Therefore, there can be no eigenfunctions on in addition to those constructed in the first step.

Case II: Eigenvalues : A basic observation is that any eigenfunction on or generalized eigenfunction on with eigenvalue can on the cylindrical part be written as , where , called the leading part, is the first mode in the -direction. is of the form

(10)

and is exponentially decreasing in .

  • First step: The scattering solutions have leading part of the form (10) with . Since is not decaying as , we should only cut off near , in order to obtain a small error term when constructing an approximate eigenfunction on from . Therefore, we must require to satisfy matching conditions at that make it a smooth function on the cylindrical part of . A short calculation shows that this is equivalent to the equation

    (11)

    Perturbation theory gives functions and spaces so that the solutions of this equation are and . So for these the function obtained from by cutting of near satisfies the eigenvalue equation on with a very small error. Therefore, one gets eigenvalues as in (6).

  • Second step: It remains to show that for large all eigenvalues on are obtained in this way. This is the theoretically most demanding part of the proof. Let be an eigenfunction of with eigenvalue . An argument directly analogous to the case won’t work since it only yields that has nonempty spectrum near , which we know anyway. Rather, we need to show that is very close to some with satisfying (11) and with very close to . The closeness of would follow from closeness of the leading parts since they control the full function, and this is proved in two steps: First, we prove an elliptic estimate which reflects the essentials of scattering theory in a compact elliptic problem and implies that eigensolutions whose part decays exponentially for (as does) are close to scattering solutions (whose part decays exponentially for ) with the same eigenvalue, see Lemmas 6 and 6. This is a stable version of the existence and uniqueness of scattering solutions on . So we obtain some very close to . In a second step it is shown that must be very close to a solution of (11): Since satisfies the matching conditions at and is very small, almost satisfies these conditions, so is very small; therefore, all that is needed is a stable version of the analysis of (11) (which, however, is quite non-trivial).

Case III: Eigenvalues : While the first step (construction of approximate eigenfunction) presents no new difficulties, the second step (proof that all eigenfunctions are obtained) is quite delicate. One difficulty is that the representation (10) is not valid at (it does not give all scattering solutions). There is a straight-forward replacement, however. More serious is showing that any eigenvalue must actually be in . This again requires a delicate stability analysis of the matching condition.

Special care needs to be taken when the multiplicities are not equal to one, since it is not enough just to construct eigenvalues, one also needs sufficiently many. A useful tool here is the notion of distance between subspaces of a vector space which we recall in Section 3.4.

If one is only interested in the case of fixed as in Theorem 1 then some of the proofs can be simplified considerably since the functions remain separated for different then. See, for example, Corollary 86. In particular the proof of Lemma 133 simplifies considerably in this case, and Theorem 9 is not needed.

1.3. Outline of the paper

In Section 2 we introduce the setup precisely and some notation. Section 3 introduces the basic analytic tools, most importantly separation of variables and distance between subspaces, as well as a basic spectral approximation lemma. In Section 4 we recall the facts from scattering theory that we need. In Section 5 we analyze which scattering solutions satisfy the matching conditions, in particular, equation (11). The basic elliptic estimate and some consequences are proven in Section 6. Theorem 1 and improvements of it are proven in Section 7, and Theorem 1.1 in Section 8, where we also discuss some special cases.

In the Appendix we collect some basic results on one-parameter families of unitary operators which are needed in the analysis of (11). Their proofs can be found in [7].

1.4. Related work

As already mentioned, the Neumann problem was treated in [4], [6], [15], [21], [5]. For Dirichlet boundary conditions, Post [20] derived the first two terms of (2) in the case of ’small’ vertex neighborhoods, see Theorem 8. In the recent preprint [17] Molchanov and Vainberg study the Dirichlet problem and show that, in the context of Theorem 1, the converge to eigenvalues of the quantum graph described in Theorem 1.1; this was conjectured in [16], where also some results on the scattering theory on non-compact graphs are obtained. However, their statements are unclear as to whether the multiplicities coincide; also, they do not consider the effect of eigenvalues on or uniform asymptotics for large . In [1] a related model is considered. The method in the previously cited papers is to compare quadratic forms or to show resolvent convergence of some sort, and in all cases only the leading asymptotic behavior is obtained.

Problems of the same basic analytic structure (cylindrical neck stretching to infinity, attached to fixed compact ends; usually with consisting of one edge only) were studied by various authors in the context of global analysis, where they occur in a method to prove gluing formulas for spectral invariants. In their study of analytic torsion (related to the determinant of the Laplacian), Hassell, Mazzeo and Melrose [10] gave a very precise description of the resolvent (in the case of closed manifolds, i.e. no boundary, but admitting edge neighborhoods which are not precisely cylindrical, just asymptotically), including its full asymptotic behavior as , using ideas of R. Melrose’s ’b-calculus’, a refined version of the pseudodifferential calculus. More direct approaches were used by Cappell, Lee and Miller [3] and by Müller [18] in the study of the -invariant (for the Dirac operator instead of the Laplacian) and by Park and Wojciechowski [19]. The author and Jerison [8] prove a special case of Theorem 1 (where is a plane domain obtained by attaching a long rectangle to a fixed domain, which is required to have width at most the width of the rectangle) by a different method (matched asymptotic expansions) and use it to prove a result about nodal lines of eigenfunctions; for this, one needs to know the asymptotic behavior to second order (i.e. one order more than written explicitly in (2)).

In the context of this literature, the main purpose of the present paper is to give a mathematically rigorous yet straightforward derivation of the limiting problem on the graph, allowing various boundary conditions, in a way that admits generalization to similar problems. For example, there are straightforward generalizations to higher order operators, systems and Schrödinger operators with potential, as long as one has a product type structure along the edges. We use existence of the scattering matrix for manifolds with cylindrical ends as a ’black box’. The main technical problem is the proof that all eigenfunctions are obtained by the given construction; here the neighborhood of the threshold requires a special effort (this is sometimes referred to as the problem with ’very small eigenvalues’ since the eigenvalues are exponentially close to ).

Notation: As usual constants may have different values at each occurence (unless otherwise stated). They depend on the data (the graph, the edge lengths, the compact manifolds), but not on . The scalar product on a space is denoted by and the norm by .

2. The setup: Combinatorial and geometric data

The following data are given:

  • Combinatorial data:

    A finite graph , with vertex set , edge set . Loops and multiple edges are allowed. Thus, may be thought of as a multiset of unordered pairs of vertices. If the vertex is adjacent to an edge , we write . A half-edge is a pair with , and we denote by the set of half edges, except that for a loop at a vertex the element appears twice in (so formally is really a multiset). Sometimes we denote a half-edge by if it arises from the edge . The neighborhood of a vertex is the (multi-)set of half-edges incident to it.

    It will be useful to think of as the disjoint union of the vertex neighborhoods, with the ’ends’ of the half-edges glued together appropriately. The glueing may be encoded by a map , where

    (12)
    (13)

    if the edge connects and (for a loop , maps one copy of to the other). is an involution, that is , and has no fixed points.

    We also assume that a positive number is given for each edge , to be thought of as half the length of . Denote the shortest half edge length by

  • Geometric data:

    • To each vertex a compact Riemannian manifold with piecewise smooth boundary1, of dimension

    • to each edge a compact Riemannian manifold with smooth boundary, of dimension ; for a half edge corresponding to an edge , we set ;

    • to each half-edge an isometry (gluing map) from to a subset of the boundary of ; we assume that is of product type near , see below; also, for each the sets are assumed to be disjoint (no overlaps of different edges);

    • partitions of the boundary of each and of the part of the boundary of each which is not in the image of any of these isometries, into two pieces denoted by indices D and N (for Dirichlet and Neumann boundary conditions); we assume sufficient regularity of this decomposition so that the boundary value problems formulated below (before (16)) are well-posed.

      More generally, one may give a pair of non-negative functions (of sufficient regularity to make the problem below well-posed) on the boundary of each (outside the gluing part) and each , with never vanishing simultaneously, to define Robin boundary conditions, see below. The D/N decomposition corresponds to being characteristic functions of a partition of these boundaries into two parts.

From this data, we define Riemannian manifolds , with piecewise smooth boundary, as follows: First, for each half-edge we attach a cylinder with cross section ,

to , using the isometry . Thus, for each vertex we get a manifold with piecewise smooth boundary

where the quotient means that each is identified isometrically with a subset of the boundary of .

On we put the cylindrical Riemannian metric

(14)

By assumption, is of product type near , which means that a neighborhood of in is isometric to with the product metric, for some . This ensures that the metrics on and define a smooth Riemannian metric on (it also fixes the smooth structure on ).

Let

(disjoint union) and denote by

the cross section resp. the cylindrical part of .

For finite the pieces are now glued together as prescribed by the graph to give the -neighborhood, , of . More precisely, induces bijections, also denoted , (since both of these cross sections are just copies of the same ), and therefore for each , and then

(15)

where is analogous to , except that is replaced by . In other words, is the union of the and cylinders for each edge of , glued together according to the structure of . (But our coordinate will run between and from both ends of the interval.) We also write

Clearly, the D/N decomposition of the boundaries of each and of each gives a corresponding decomposition of the boundary of each , of and , and . The Riemannian metrics define Laplace operators on these spaces, for which we impose Dirichlet boundary conditions on the D part of the boundary and Neumann boundary conditions on the N part (but no boundary condition at the boundary piece of ). More generally, one may consider Robin boundary conditions, , where the functions on (etc.) are induced by those on the boundaries of , , by making them independent of the cylinder coordinate .

By (14),

(16)
{remark}

The Riemannian metric (14) expresses the fact that the cylindrical part has length . Instead, one could use the coordinate , then one would get the more standard form

(17)

and on . The -coordinate is more convenient for some calculations, but in the -coordinate is given by the same range of , , for all . We switch between these coordinates as is convenient.

3. Basics of the analysis

3.1. Notation

Since is compact, the Laplacian has compact resolvent and therefore discrete spectrum. Let be the eigenvalues of , with finite dimensional eigenspaces . We also write

Let be the orthogonal projection to in , and .

Let . We decompose any in its ’vertical’ and components, over the cylindrical part :

where and similarly for . Here and throughout the paper we identify functions on with functions on whose values are functions on . In particular,

Since and acts on each separately, we have

(18)

We write elements as with . If then .

The data define two important linear maps which restrict to linear maps :

The map is diagonal with respect to the splitting (18) and encodes the edge lengths

(19)

The map is defined by the involution encoding the graph structure:

(20)

We denote

(21)

Since is a self-adjoint involution, the decomposition is orthogonal. For we denote the corresponding decomposition .

By (16) and (17), we have

(22)
(23)

with respect to the decomposition .

3.2. Matching conditions at

We regard functions on as functions on satisfying suitable matching conditions at . For solutions of the eigenfunction equation we get: {lemma} Let be an eigenfunction of with eigenvalue .

Then defines an eigenfunction of if and only if extends smoothly to and

(24)

The upper refer to the -decomposition (21). For let

(25)

Then if and only if satisfies the matching conditions at . The particular scaling in (25) is motivated by the calculation (55).

Proof.

This is just the fact that the solution to a second order elliptic partial differential equation can be continued across a hypersurface, as a solution, if and only if at the hypersurface it is continuous and its normal derivatives match. Set , . Then and , so (24) means , for all , that is, continuity of and of at in . ∎

For a function on , when we write we always assume that extends smoothly to .

3.3. Separation of variables

The following simple lemma is basic to all the analysis.

{lemma}

Let . Let satisfy .

  • The leading part of has the form

    (26)
    (27)
    (28)

    with . (Replace by to express this in terms of .)

  • If then

    (29)
  • in (a) and in (b) are uniquely determined by . If and is polynomially bounded as then in (26) and for all in (29).

  • Assume and extends to a solution on , or and is polynomially bounded. Then there is a constant such that for all we have

    (30)
    if , and
    (31)

    if .

  • If and is an eigenfunction on then, for the representation in (29),

    (32)

    if for any .

Proof.

For let be the projection and let . By (23), satisfies the differential equation

(33)

and this has the solutions given by the formulas in (a) and by the summands in (b). Clearly, and therefore are determined by , and if is polynomially bounded then so is for each , so (c) follows. This immediately gives (d) in the case . If and extends to a solution on then may be regarded as function . We now use the coordinate . If then one can express the solution of (33) by its values at , for any : Write , then

For any and one obtains easily at :

Summing over in the case yields the estimate on in (30). The estimate on the derivative and estimate (31) are obtained similarly.

(e) follows similarly to (d), using , and . ∎

It is clear from the proof that the behavior of solutions is different for . For the purpose of this paper, we will always consider . Define by