Quantum graphs on radially symmetric antitrees

We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,\infty)$.


Introduction
This paper is devoted to one particular class of infinite quantum graphs, namely Kirchhoff Laplacians on radially symmetric antitrees. Antitrees appear in the study of discrete Laplacians on graphs at least since the 1980's (see [12] and also [11,Section 2]) and they attracted a considerable attention after the work of Wojciechowski [48]. More precisely, Wojciechowski used them in [48] (see also [30, §6] and [23]) to construct graphs of polynomial volume growth for which the (discrete) physical Laplacian is stochastically incomplete and the bottom of the essential spectrum is strictly positive, which is in sharp contrast to the manifold setting (cf. [9], [21], [22]). These apparent discrepancies were resolved later using the notion of intrinsic metrics, with antitrees appearing as key examples for certain thresholds (see [18,24,25,29]). During the recent years, antitrees were also actively studied from other perspectives and we only refer to a brief selection of articles [1], [8], [11], [20], [43], where further references can be found.
In this paper, we consider antitrees from the perspective of quantum graphs and perform a detailed spectral analysis of the Kirchhoff Laplacian on radially symmetric antitrees. Our discussion includes characterization of self-adjointness and a complete description of self-adjoint extensions, spectral gap estimates and spectral types (discrete, singular and absolutely continuous spectrum).
Before proceeding further, let us first recall necessary definitions. Let G d = (V, E) be a connected, simple (no loops or multiple edges) combinatorial graph. Fix a root vertex o ∈ V and then order the graph with respect to the combinatorial spheres S n , n ∈ Z ≥0 (notice that S 0 = {o}). Definition 1.1. A connected simple rooted (infinite) graph G d is called an antitree if every vertex in S n , n ≥ 1 1 , is connected to all vertices in S n−1 and S n+1 and no vertices in S k for all |k − n| = 1.
Notice that combinatorial antitrees admit radial symmetry and every antitree is uniquely determined by its sphere numbers s n = #S n , n ≥ 0 (see Figure 1).
If every edge of G d is assigned a length |e| ∈ (0, ∞), then G = (G d , | · |) is called a metric graph. Upon identifying each edge e with the interval of length |e|, G may be considered as a "network" of intervals glued together at the vertices. In the following we shall denote combinatorial and metric antitrees by A d and, respectively, A. The analogue of the Laplace-Beltrami operator for metric graphs is the Kirchhoff Laplacian H (or Kirchhoff-Neumann Laplacian, see Section 3.1), also called a quantum graph. It acts as an edgewise (negative) second derivative f e → − d 2 dx 2 e f e , e ∈ E, and is defined on edgewise H 2 -functions satisfying continuity and Kirchhoff conditions at the vertices (we refer to [2,3,15,17,32,40] for more information and references).
Our approach employs the high degree of symmetry and this naturally demands symmetry assumptions also on the choice of edge lengths (Remark 10.4(i) shows this is indeed necessary for our results): Hypothesis 1.1. We shall assume that the metric antitree A is radially symmetric, that is, for each n ≥ 0, all edges connecting combinatorial spheres S n and S n+1 have the same length, say ℓ n > 0. One of our main motivations is Lemma 8.9 in [32]. More precisely, the symmetry of antitrees structure turned out useful in studying isoperimetric estimates and we were even able to compute explicitly the bottom of the essential spectrum of some (non-equilateral) quantum graphs (see [32, §8.2]). Despite an enormous interest in quantum graphs during the last two decades, to the best of our knowledge a detailed discussion of their spectral properties without further restrictions on edges lengths (for instance, one of the most common assumptions is inf e∈E |e| > 0) has so far been obtained only for a few models and the most studied ones are radially symmetric trees (see e.g. [6,10,16,37,38,45]). However, the assumption that G is a tree prevents many interesting phenomena to happen (for instance, by [32,Lemma 8.1], in this case the Kirchhoff Laplacian, actually, its Friedrichs extension, is boundedly invertible if and only if sup e∈E |e| < ∞; in fact, this condition is only necessary in general [44]). Hence our goal in this work is to present a model which can be thoroughly analyzed but still exhibits in some sense rich spectral behavior.
Let us now briefly describe the content of the paper and our main results. To some extent we follow the approach developed by Naimark and Solomyak for radially symmetric trees (see [37,38] and also [10,44,45]) and use some ideas from [8], where discrete Laplacians on radially symmetric "weighted" graphs have been analyzed. However, some modifications are necessary since comparing to [10,38,45] we are dealing with a completely different class of graphs (antitrees have a lot of cycles) and, in contrast to discrete Laplacians [8], we have to deal with unbounded operators (even when restricting to compact subsets of a metric graph) and in this case a search for reducing subspaces is a rather complicated task 2 First of all, the radial symmetry of A naturally hints to consider the space F sym of radially symmetric functions (w.r.t. the root o ∈ V). It turns out that F sym is indeed reducing for the pre-minimal Kirchhoff Laplacian H 0 (this means that H 0 as well as its closure H = H 0 , the minimal Kirchhoff Laplacian, commutes with the orthogonal projection onto F sym ) and its restriction H 0 ↾ F sym is unitarily equivalent to a pre-minimal Sturm-Liouville operator H 0 defined in L 2 ((0, L); µ) 2 After the submission of our paper we learned about the preprint [7] dealing with a similar decomposition in the general case of family preserving metric graphs, which includes antitrees as a particular example. However, the main focus of [7] is on the existence of a decomposition in a rather general situation, whereas in our work we use it mainly as a starting point for the spectral analysis.
by the differential expression and subject to the Neumann boundary condition at x = 0. Here t 0 = 0, t n = k≤n−1 ℓ k for all n ≥ 1 and L = n≥0 ℓ n (see Section 3.2). Moreover, the remaining part of H = H 0 decomposes into an infinite sum of self-adjoint (regular) Sturm-Liouville operators (see Theorem 3.5; its proof is given in Sections 2 and 3). This decomposition is the starting point of our analysis since it enables us to investigate H using the well-developed spectral theory of Sturm-Liouville operators. For example, this immediately provides a self-adjointness criterion together with a complete description of self-adjoint extensions of H (see Section 4). Namely, since all the summands in (3.18) except H = H 0 are self-adjoint operators, we reduce the problem to the study of the operator H 0 . Employing Weyl's limit point/limit circle classification, we obtain in Theorem 4.1 that deficiency indices of H are at most 1. Moreover, H is self-adjoint if and only if A has infinite total volume, i.e.

vol(A) :=
If A has finite total volume, vol(A) < ∞, all self-adjoint extensions can be described through a single boundary condition (in particular, this also provides a description of the domain of the Friedrichs extension). Moreover, all of their spectra are purely discrete and eigenvalues satisfy Weyl's law (see Corollary 5.1).
If vol(A) = ∞, i.e., H is self-adjoint, it was already observed in [32, Section 8.2] that σ(H) is not necessarily discrete. In Section 5, we characterize the cases when H has purely discrete spectrum and when its resolvent H −1 belongs to the trace class (see Theorem 5.3 and Theorem 5.5). Let us stress that our main tool is the spectral theory of Krein strings [27] (see also [13]). More precisely, by a simple change of variables H can be transformed into the string form (see (5.9)) and then one simply needs to use the corresponding results from [26,27]. Section 6 is devoted to spectral estimates, i.e., the investigation of the bottom of the spectrum λ 0 (H) of H, λ 0 (H) := inf σ(H). This can be solved again by using the results of Kac and Krein from [26]. More precisely, we characterize the positivity of λ 0 (H) (Theorem 6.1 and Theorem 6.3) and derive two-sided estimates (Remark 6.2). Let us also mention at this point that the decomposition (3.18) indicates the way to compute the isoperimetric constant of a radially symmetric antitree (see Theorem 7.1) and hence it is interesting to compare Theorem 6.1 and Theorem 6.3 with the estimates obtained recently in [32] (see Remark 7.2).
To our best knowledge, the theory of Krein strings is applied in the context of quantum graphs for the first time. In fact, most of the analysis in Sections 5 and 6 can be performed with the help of Muckenhoupt inequalities [36] since the questions addressed in these sections allow a variational reformulation (in particular, Solomyak used this approach in [45] to investigate quantum graphs on radially symmetric trees). However, spectral theory of strings enables us to treat more subtle problems (like the study of the structure of the essential spectrum of H). In Section 9, we employ the recent results from [4] and [14] on the absolutely continuous spectrum of strings to construct several classes of antitrees with absolutely continuous Theorem 9.6). Notice that to prove this claim we employ the analog of the Szegő theorem for strings recently established by Bessonov and Denisov [4]. Antitrees with polynomially growing sphere numbers satisfy the last assumption, however, it can be shown that in this case the usual trace class arguments do not apply (see Remark 9.4). Let us also emphasize that similar to the case of trees quantum graphs typically have purely singular spectrum in the case of antitrees (see Section 8). However, to the best of our knowledge, the only known examples of quantum graphs on trees having nonempty absolutely continuous spectrum are eventually periodic radially symmetric trees (see [16,Theorem 5.1]).
In the final section we demonstrate our results by considering two special classes of antitrees and complement the results of [32,Section 8.2]. In Section 10.1 we consider antitrees with exponentially increasing sphere numbers and demonstrate that in this case there are a lot of similarities with the spectral properties of quantum graphs on radially symmetric trees. Antitrees with polynomially increasing sphere numbers are treated in Section 10.2 and this class of quantum graphs exhibits a number of interesting phenomena. For example, one can show a transition from absolutely continuous spectrum supported on [0, ∞) to purely discrete spectrum (see Corollary 10.7).
In Appendix A we discuss the eigenvalues of a special class of regular Sturm-Liouville operators and in Appendix B we collect several examples of antitrees whose degree function takes finitely many values and the absolutely continuous spectrum of the corresponding Laplacian is [0, ∞).
Finally, let us stress that our approach based on spectral theory of Krein strings enables us (without almost no effort) to extend most of the results obtained in this paper to arbitrary second order differential operators (namely, one can replace the second derivative by a weighted Sturm-Liouville operator − 1 r(x) d dx p(x) d dx or even by a string differential expression − d 2 dω(x)dx ), of course keeping the radial symmetry assumption on the coefficients.
2. Decomposition of L 2 (A) 2.1. Auxiliary subspaces. Let A be a metric radially symmetric antitree with sphere numbers {s n } n≥0 and lengths {ℓ n } n≥0 . Upon identifying every edge e with a copy of the interval I e = [0, |e|] and considering A as the union of all edges glued together at certain endpoints, one can introduce the Hilbert space L 2 (A) of functions f : A → C as L 2 (A) = ⊕ e L 2 (e). Next, denote and let H n := C snsn+1 , n ≥ 0. Notice that s n s n+1 is the number of edges in E + n , where E + n is the set of edges connecting S n with S n+1 . Enumerating the vertices in each sphere, let each entry a ij of some a = (a ij ) i,j ∈ H n correspond to a coefficient of the edge e ∈ E + n connecting the i-th vertex of S n with the j-th vertex of S n+1 .
Moreover, we can identify each function f : A → C in a natural way with the sequence of functions f = (f n ) n≥0 such that f n : I n → H n . In fact, f n is given by where x ij (t) is the unique x ∈ A, such that |x| = t and x lies on the edge connecting the i-th vertex in S n with the j-th vertex of S n+1 . Notice that the map is an isometric isomorphism since for all f, g ∈ L 2 (A). Next we introduce the following subspaces: H sym n := a ∈ H n | a ij = a 11 ∀i, j , It is straightforward to check that the above spaces are mutually orthogonal and their dimensions are given by Notice also that if s n = 1 for some n ≥ 1, then H + n = H 0 n = H 0 n−1 = H − n−1 = {0}. One can also describe the above subspaces by identifying H n with the tensor product C sn ⊗ C sn+1 . For example, setting for all n ≥ 0, we get H sym n = span{1 n }. Notice that {a j sn } sn j=1 forms an orthogonal basis in C sn for all n ≥ 0. In particular, a sn sn = 1 sn and a j sn 2 = s n . Hence setting where 1 ≤ i ≤ s n and 1 ≤ j ≤ s n+1 , we easily get (2.9) Finally, observe that a i,j n 2 = s n s n+1 (2.10) for all 1 ≤ i ≤ s n , 1 ≤ j ≤ s n+1 and n ≥ 0.

2.2.
Definition of the subspaces. The decomposition (2.4) naturally induces a decomposition of the Hilbert space L 2 (A). First consider the subspace Clearly, it consists of functions which depend only on the distance to the root: Moreover, its orthogonal complement is given by Next we need to decompose F ⊥ sym . Set F 0 n := {f ∈ L 2 (A)| f n : I n → H 0 n ; f k ≡ 0, k = n} (2.14) for all n ≥ 1. Taking into account the definition of H 0 n , it is not difficult to see that Here, for every v ∈ V, E + v and E − v denote the edges connecting v with the next and, respectively, previous combinatorial spheres.
We need to be more careful with the remaining part since our aim is to find reducing subspaces for the quantum graph operator H. For every v ∈ V \ o, define the subspace F v consisting of functions which vanish away of E v , where E v is the set of edges emanating from v. Moreover, on the corresponding star E v they depend only on the distance to the root, that is, (2.15) Notice that F v and F u are orthogonal for u = v if u and v are not adjacent vertices. Next for all n ≥ 1 consider the spaces and Notice that with respect to the decomposition (2.4), we have Thus, we arrive at the following result.
Lemma 2.1. The Hilbert space L 2 (A) admits the decomposition Proof. The orthogonality of the subspaces in (2.19) follows directly from (2.3) and (2.4). Hence we only need to show that every f ∈ L 2 (A) is contained in the right hand side of (2.19). Since L 2 (A) = ⊕ e∈E L 2 (e), it suffices to prove this claim in the case when f is zero except on a single edge e ∈ E. Suppose that e ∈ E + n for some n ≥ 0. Then for almost every t ∈ I n we have where P j n : H n → H j n is the orthogonal projection in H n onto H j n , j ∈ {sym, +, −, 0}. Define f j : A → C as the function identified with the sequence of functions f j = (f k j ) k≥0 given by for a.e. t ∈ I k . Then f j ∈ L 2 (A) for all j ∈ {sym, +, −, 0} and Since f k j (t) ∈ H j k for a.e. t ∈ I k , we conclude that f sym ∈ F sym , f 0 ∈ F 0 n , f + ∈ F n and f − ∈ F n+1 . Our next aim is to write down explicit formulas for projections onto the subspaces in the decomposition (2.19). First, for anyf ∈ L 2 (I n ) and a ∈ H n , we setf :=f ⊗a.
Recalling that every function f : A → C can be identified via (2.2) with the sequence of vector-valued functions f = (f n ) n≥0 , we denote Note that the orthogonal projection P n a of L 2 (A) onto F n a is given by where U is the isometric isomorphism (2.2). Combining the form of P n a with the decomposition (2.4) and (2.6), (2.9), we easily obtain the following result. Lemma 2.2. Let 1 n ∈ H n and a i,j n ∈ H n , n ≥ 0 be given by (2.5) and (2.8). Then the orthogonal projections in the decomposition (2.19) are given by n ≥ 1, (2.23)

Reduction of the quantum graph operator
In this section, we show that each of the spaces in the above decomposition (2.19) is reducing for the quantum graph operator with Kirchhoff conditions and also obtain a description of the corresponding restrictions.

3.1.
Kirchhoff 's Laplacian. Let us briefly recall the definition of the Laplacian on a metric graph (for details we refer to [3,17,32]). Let L 2 (A) be the corresponding Hilbert space and the subspace of compactly supported L 2 -functions will be denoted by are well defined for all f ∈ H 2 (A \ V) and every vertex v ∈ V. Imposing these boundary conditions and restricting to compactly supported functions we get the pre-minimal operator H 0 acting edgewise as the (negative) second derivative The operator H 0 is symmetric and its closure H = H 0 is called the minimal Kirchhoff Laplacian. First, we need the following simple but useful fact.
and only if f = (f n ) n≥0 satisfies the following conditions: Proof. The proof is straightforward. We only need to mention that (i) is equivalent to the fact that f is compactly supported; (ii) means that f belongs to the Sobolev space H 2 on each edge e ∈ E; (iii) and (iv) are continuity and Kirchhoff's conditions at the vertices.
3.2. The subspace F sym . Set I L = [0, L), and define the length L and the weight function µ : Consider the (pre-minimal) Sturm-Liouville operator H 0 defined in L 2 (I L ; µ) by the differential expression (3.6) More concretely, H 0 acts as a negative second derivative and its domain dom(H 0 ) consists of functions f ∈ L 2 (I L ; µ) having compact support in I L , belonging to H 2 on every interval I n and at each point t n satisfying the boundary conditions f is continuous at t n , Here we set s −1 := 0 in the case n = 0 for notational simplicity and the corresponding condition (3.7) reads as the Neumann boundary condition at t = 0.
Lemma 3.2. The subspace F sym reduces the operator H 0 . Moreover, its restriction H 0 ↾ F sym onto F sym is unitarily equivalent to the operator H 0 .
Proof. First let us show that f sym : Clearly, by continuity of f and (2.21), (2.22), f sym satisfies (i) and (ii). Moreover, both (f sym ) n i,j (t n +) and (f sym ) n k,m (t n+1 −) depend only on n ≥ 0. Since f satisfies both (iii) and (iv), we obtain that (f sym ) 0 1,j (0+) does not depend on j and which holds for all i ∈ {1, . . . , s n }, n ≥ 1. Moreover, for n = 0 we have Noting that H 0 is symmetric and F sym is clearly invariant for H 0 we conclude that F sym is reducing for H 0 .
To prove the last claim, observe that the subspace F sym is isometrically isomorphic to the Hilbert space L 2 (I L ; µ). Indeed, for every f ∈ F sym , set where x e (t) is the unique point on e satisfying |x e (t)| = t. Consider the map Clearly, for every f ∈ F sym , f n (t) =f (t) ⊗ 1 n for a.e. t ∈ I n and hence

It turns out that
Indeed, H 0 acts as the negative second derivative on every edge e ∈ E and hence for every f ∈ F sym we get for all n ≥ 0. Therefore, it remains to show that U s (F sym ∩ dom(H 0 )) = dom(H 0 ). In fact, we only need to show that everyf = U s f with f ∈ F sym satisfies (3.7) if and only if f ∈ dom(H 0 ). Indeed, by (3.9) and continuity of f , . This finishes the proof of Lemma 3.2.

3.3.
Restriction to F 0 n . Our next aim is to show that each F 0 n , n ≥ 1, is a reducing subspace for H 0 and its restriction is unitarily equivalent to (s n −1)(s n+1 −1) copies of h n , the second derivative with the Dirichlet boundary conditions on L 2 (I n ), By Lemma 2.2, this will be a consequence of the following lemma.
Lemma 3.3. Let n ≥ 1 be such that s n > 1 and s n+1 > 1. Then each of the subspaces F n a , where a = a i,j n with 1 ≤ i < s n and 1 ≤ j < s n+1 , is reducing for the operator H 0 . The restricted operator H 0 ↾ F n a is unitarily equivalent to the operator h n defined by (3.12).
Proof. Clearly, F n a is invariant for H 0 . Since H 0 is symmetric, we only have to prove thatf := P n a f ∈ dom(H 0 ) whenever f ∈ dom(H 0 ). In fact, we need to show that f := U (P n a f ) given by (2.21) satisfies conditions (i)-(iv) of Lemma 3.1. Conditions (i) and (ii) are obviously satisfied since f ∈ dom(H 0 ) and by the definition of U (P n a f ). Sincef m = 0 for all m = n and n ≥ 1, (iii) clearly holds and, moreover, we need to verify (iv) only at t n and t n+1 .
Let us start with continuity. Suppose a = a i,j n for some 1 ≤ i < s n and 1 ≤ j < s n+1 . First observe thatf Here we employed the continuity of f , f n k,j (t n +) = f n k,1 (t n +) for all j ∈ {1, . . . , s n+1 }, together with (2.8). This shows thatf satisfies the first condition in (iv).
Next observe that for all k ∈ {1, . . . , s n }. Since (f n−1 ) ′ = 0,f satisfies (iv) at t n . Similar arguments shows that (iv) holds true at t n+1 as well. This finishes the proof of the inclusioñ establishes an isometric isomorphism of L 2 (I n ) onto F n a , it is straightforward to verify the last claim and we left it to the reader.
3.4. Restriction to F n . Next, we show that F n , n ≥ 1 is reducing for H 0 as well and the corresponding restriction is unitarily equivalent to s n − 1 copies of the operator h n defined by on L 2 ((t n−1 , t n+1 ); µ) and equipped with Dirichlet conditions at the endpoints. Here the weight function µ is defined by (3.4). The domain of h n admits a very simple description since inside I n−1 and I n the differential expression τ n reduces to the negative second derivative and hence dom( h n ) consists of functions which are H 2 in I n−1 and I n , satisfy the Dirichlet conditions at t n−1 and t n+1 and also the following coupling conditions at t n : (3.14) Recall that F n = ran(P n ), where the projection P n is given by (2.24). By (2.8) and (2.5), a sn−1,j n−1 Denoting the summands in (3.15) by P j n , j ∈ {1, . . . , s n − 1}, we set F j n := ran( P j n ) = F n−1 Since F n = sn−1 j=1 F j n , these claims will follow from the following lemma: Lemma 3.4. Every subspace F j n with n ≥ 1 and j ∈ {1, . . . , s n − 1}, is reducing for the operator H 0 . Moreover, its restriction onto F j n is unitarily equivalent to h n . Proof. Since F j n is invariant for H 0 and H 0 is symmetric, we only need to show that for every f ∈ dom(H 0 ) its projectionf := P j n f onto F j n also belongs to dom(H 0 ). Following step by step the proof of Lemma 3.3, we only need to show thatf := Uf satisfies condition (iv) of Lemma 3.1 at t n .
First observe that by (2.21) Notice that However, by Lemma 3.1, and hence we get for all k ∈ {1, . . . , s n }. This shows thatf satisfies the first equality in condition (iv) of Lemma 3.1. Let us check the second one. However, we have This shows thatf satisfies all the conditions of Lemma 3.1 and hencef ∈ dom(H 0 ). Finally, it is straightforward to check that the map U j n :

18)
where H = H 0 and the operators H 0 , h n and h n are defined in Sections 3.2, 3.3 and 3.4, respectively.

Self-adjointness
Theorem 3.5 reduces the spectral analysis of quantum graph operators on radially symmetric antitrees to the analysis of certain classes of Sturm-Liouville operators. Moreover, the Sturm-Liouville operators h n and h n in the decomposition (3.18) are self-adjoint for all n ≥ 1 and their spectra can be computed explicitly. This enables us to perform a rather detailed study of spectral properties of the operator H = H 0 . We begin with the characterization of self-adjoint extensions of the operator H.
The operators P sym and U s are given, respectively, by (2.22) and (3.10) and Proof. (i) By Theorem 3.5, the operator H is self-adjoint only if so are the operators on the right-hand side in the decomposition (3.18). However, both h n and h n are self-adjoint for all n ≥ 1. The self-adjointness criterion for H = H 0 follows from the standard limit point/limit circle classification (see, e.g., [47]). Namely, the equation τ y = 0 with τ given by (3.5), has two linearly independent solutions converges. Since s n s n+1 ≥ 1 for all n ≥ 0, this series converges exactly when the series in (4.1) converges. The Weyl alternative finishes the proof of (i). exist and are finite (see, e.g., [47]).

Discreteness
As an immediate corollary of Theorem 4.1 we obtain the following result.
Corollary 5.1. If vol(A) < ∞, then the spectrum of each self-adjoint extension H θ of H is purely discrete and, moreover, for all θ ∈ [0, π).
Here N (λ; A) is the eigenvalue counting function of a (bounded from below) self-adjoint operator A with purely discrete spectrum. Namely, where {λ k (A)} k≥0 are the eigenvalues of A (counting multiplicities) ordered in the increasing order.
Proof. By Theorem 3.5, Since s n ≥ 1 for all n ≥ 1, vol(A) < ∞ implies that ℓ n = o(1) as n → ∞ and hence both sets ∪ n≥1 σ(h n ) and ∪ n≥1 σ( h n ) have no finite accumulation points. It remains to note that the spectrum of H(θ) is discrete in this case as well.
However, we are not aware (except a few special cases) of a closed form of eigenvalues of h n . It is not difficult to show that σ( h n ) consists of simple positive eigenvalues { λ k } k≥1 satisfying (5.3) and even to express σ( h n ) with the help of the arctangent function with two arguments, (see Appendix A)although this does not lead to a closed formula.
In the case vol(A) = ∞, the spectrum of H may have a rather complicated structure. In particular, it may not be purely discrete. The next result provides a criterion for H to have purely discrete spectrum. Set Proof. Denote By Theorem 4.1(i), H is self-adjoint and hence (3.18) implies that x 0 ds µ(s) ) and then to apply the Kac-Krein criterion [26]. To be more precise, it is straightforward to verify that H is unitarily equivalent to the operator H defined in the Hilbert space L 2 ([0, L µ ); µ) by the differential expression for all x ∈ [t n , t n+1 ) and hence sufficiency of (5.6) follows. Moreover, straightforward calculations show that which implies the necessity of (5.6). Notice also that the right-hand side in the last inequality is strictly greater than 1 4 ℓ 2 n , which also implies (i). It remains to note that the spectra of the operators H 1 and H 2 are discrete if condition (i) is satisfied (see (5.4) and (5.3)).

Remark 5.4. Let us mention that in fact both conditions (i) and (ii) in Theorem 5.3 follow from (iii).
If vol(A) = ∞ and the corresponding Hamiltonian H has purely discrete spectrum, it follows from the proof of Weyl's law (5.1) that N (λ;H) √ λ → ∞ as λ → ∞. However, we can characterize radially symmetric antitress such that the resolvent of the corresponding quantum graph operator H belongs to the trace class.
Similarly, the spectrum of the self-adjoint operator h n also consists of simple positive eigenvalues, however, we are not aware of their closed form. Instead one can employ the standard Dirichlet-Neumann bracketing, that is, to estimate the eigenvalues of h n via the eigenvalues of the operators h D n and h N n subject to Dirichlet, respectively, Neumann boundary conditions at t n : for all k ≥ 1. Thus, we get Using the Dirichlet eigenvalues, one can prove a similar bound from below. Moreover, combining the latter with (5.15) implies that the resolvents of both H 1 and H 2 belong to the trace class exactly when n≥1 (s n s n+1 − 1)ℓ 2 n < ∞. Notice that the latter in particular shows that {ℓ n } n≥0 ∈ ℓ 2 and combining this fact with (5.18) we arrive at (5.15). This completes the proof of Theorem 5.5.
Remark 5.6. Using the same arguments and the results from [28,42] one would be able to characterize radially symmetric antitrees such that the resolvent of the corresponding Kirchhoff Laplacian belongs to the Schatten-von Neumann ideal S p , p ∈ (1, ∞) (and even to other trace ideals), however, these results look cumbersome and we decided not to include them.

Spectral gap estimates
We restrict our discussion to the case vol(A) = ∞ for several reasons. Of course, the main one is the fact that in this case H 0 is essentially self-adjoint and this simplifies some considerations. However, for finite volume metric graphs the corresponding estimates remain to be true for the Friedrichs extension of H 0 .
Our next goal is to estimate the bottom of the spectrum of the operator H.
The operator H can be studied in the framework of Krein strings, however, we need to apply the Kac-Krein criteria [26] to the dual string since both Corollary 1.1 and Remark 2.2 in [26] are stated subject to the Dirichlet boundary condition at x = 0. For a detailed discussion of dual strings we refer to [27, §12] and the desired connection is [27, equality (12.6)] 4 . More precisely, assuming that L µ < ∞ and then applying Theorem 1 from [26], we get the estimate x ∈ (0, L µ ). Taking into account [26, Remark 2.2], we conclude that the condition L µ < ∞ is also necessary for the positivity of λ 0 (H). It remains to note that the necessity of (i) follows from (iii). Indeed, assuming the converse, that is, there is a sequence of lengths ℓ n k tending to infinity, and then choosing x n k as the middle points of the corresponding intervals, one immediately concludes that C(L) = ∞ by evaluating (6.1) at x n k . (ii) One can prove Theorem 6.1 avoiding the use of the Kac-Krein results [26]. Namely, with the help of the Rayleigh quotient, one can rewrite the inequality λ 0 (H) > 0 as a variational problem and then apply Muckenhoupt's inequalities (see, e.g., [34, §1.3.1], [36]). In particular, M. Solomyak employed this approach in the study of quantum graph operators on radially symmetric trees (see [45, §5]). (iii) It is interesting to compare Theorems 6.1 and 6.3 with volume growth estimates (cf. [46]). For instance, by [32, Theorem 7.1],

Isoperimetric constant
Recall that the isoperimetric constant α(G) of a metric graph G is (see [32, §3]) where the infimum is taken over all finite connected subgraphs G = ( V, E). Here is the boundary of G and Computation of the isoperimetric constant is known to be an NP-hard problem, however, due to the presence of symmetries, we are able to find α(A) for radially symmetric antitrees. Proof. The decomposition obtained in Theorem 3.5 suggests to take the infimum in (7.1) only over radially symmetric subgraphs. Namely, choosing A n for every n ≥ 0 as the subgraph consisting of all edges between the root o and the combinatorial sphere S n+1 , we have ∂A n = S n+1 and deg An (v) = s n for all vertices v ∈ S n+1 . Hence by (7.1) we get Thus it remains to show that indeed it suffices to restrict the infimum in (7.1) to the family {A n } n≥0 . Observe that {A n } n≥0 is a net, that is, for every finite connected subgraph A of A there is n ≥ 0 such that A is a subgraph of A n . Hence we will proceed by induction in n.

Let us start with subgraphs
Take n ≥ 1 and assume that Taking into account (7.6), this proves the claim. Consider a new graph A ′ obtained from A by adding all possible edges connecting S n with S n−1 and S n+1 such that the new graph A ′ is connected. By construction, We also need another subgraph A ′′ of A obtained by removing the edges of A connecting S n+1 with S n \ ∂ A and also S n \ ∂ A with the vertices in S n−1 ∩ ∂ A.
The obtained graph A ′′ is a connected subgraph of A n−1 and hence satisfies the induction hypothesis (7.5). Our aim is to show that Now observe that if (7.7) fails to hold, then (7.9) and (7.11) would imply vol( A) , (7.12) and, moroever, (7.8) and (7.10) lead to This contradiction proves (7.7) and hence finishes the proof of (7.3).
Comparing (7.14) and (7.3) with (6.2) and (6.11), we conclude that positivity of the isoperimetric constant is indeed only sufficient for λ 0 (H) > 0. For example, α(A) = 0 whenever vol(A) = ∞ and {s n s n+1 } n≥0 has a bounded subsequence. (ii) The isoperimetric constant α(A) measures the ratio of the number of boundary points of A n to the volume of A n and thus provides a lower bound for λ 0 (H). The volume growth estimate (6.14) provides an upper bound by relating the exponential growth of the volume of A n with respect to its diameter. Notice that the volume of the subgraph A n also appears in (6.10)-(6.11).
The meaning of the other quantity in (6.11), namely, of k≥n ℓ k s k s k+1 , which however provides two-sided estimates, remains unclear to us.

Singular spectrum
Using the isometric isomorphism U µ : f → √ µf between Hilbert spaces L 2 (I L ; µ) and L 2 (I L ), it is straightforward to check that the pre-minimal operator H 0 defined in Section 3.2 is unitarily equivalent to the operator H 0 defined in L 2 (I L ) by Since µ is piece-wise constant on (0, L), the domain of H 0 consists of compactly supported functions f ∈ L 2 c (I L ) such that f ∈ H 2 (I n ) for all n ≥ 0 and also satisfying the following boundary conditions for all n ≥ 1. Denote the closure of H 0 by H. The operator H has actively been studied since its spectral properties play a crucial role in understanding spectral properties of Kirchhoff Laplacians on radial metric trees (let us only mention [6,16]). It turns out that one can immediately apply most of the results from [6] and [16] in order to prove the corresponding spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. However, we need the following assumptions on the geometry of metric antitrees:  In contrast to radially symmetric trees, antitrees always have a rather rich point spectrum (see Theorem 3.5). Moreover, under the assumptions of Hypothesis 8.1 this point spectrum is not a discrete subset, that is, it has finite accumulation points (see Remark 5.2). On the other hand, similar to [6, Theorem 7], we can construct a class of antitrees such that σ(H) is purely singular continuous. Moreover, it is possible to show that under the assumption ℓ * (A) > 0 this situation is in a certain sense typical (cf.  The proof is again omitted since it is analogous to that of [16, Theorem 5.1].

Absolutely continuous spectrum
The decomposition (3.18) shows that σ ac (H) = σ ac (H) (9.1) and both have multiplicity at most 1. The results of the previous section show that antitrees with nonempty absolutely continuous spectrum is a rare event. Our main aim in this section is to apply two recent result from [4] and [14] on the absolutely continuous spectrum of Krein and generalized indefinite strings, respectively, in order to construct several classes of antitrees with rich absolutely continuous spectra, however, which are not eventually periodic in the sense of Theorem 8.3. We begin with the following result.
Theorem 9.1. Let A be an infinite radially symmetric antitree such that Also, let µ be the function given by (3.4). If Proof. We only need to use Theorem 2 from [4]. Indeed, as we know (see the proof of Theorem 6.1), the operator H is unitarily equivalent to the Krein string operator H given by (5.9)-(5.11). Applying now Theorem 2 from [4] to the operator H, after straightforward calculations the corresponding condition (1.9) from [4] turns into (9.2).
Remark 9.2. Let us mention that in Theorem 9.1, upon suitable modifications of [4, Theorem 2], one can replace the intervals (n, n + 2) by intervals I n , n ≥ 0 which "asymptotically" behave like (n, n + 2) (actually, by intervals with lengths uniformly bounded from above as well as by a positive constant from below and satisfying a suitable overlapping property [5]), however, one has to replace 4 by a square of the length of the corresponding interval: n≥0 In Let us first demonstrate the above result by considering an example of equilateral antitrees and then we shall extend it to a much wider setting (see Theorem 9.6 below). Proof. Setting I n = (ℓn, ℓ(n + 2)), n ≥ 0, straightforward calculations show that holds exactly when the operator H considered in Section 8 is a trace class perturbation (in the resolvent sense) of the free Hamiltonian − d 2 dx 2 acting in L 2 (R + ) and hence in this case the Birman-Krein theorem implies σ ac (H) = R ≥0 . However, (9.5) does not hold already for polynomially growing equilateral antitrees, e.g., take s n = n + 1 (see also Section 10.2). Moreover, (9.4) is equivalent to the fact that H is a Hilbert-Schmidt class perturbation (in the resolvent sense) of the free Hamiltonian.
The rather strong assumption that A is equilateral can indeed be replaced by ℓ * (A) > 0. In order to do this, it will turn out useful to rewrite (9.2). Let M := ran(µ) = {s n s n+1 : n ∈ Z ≥0 } (9.6) be the image of the function µ defined in (3.4). For every s ∈ M, we set that is, I s is the preimage of {s} ∈ M with respect to µ.
Lemma 9.5. Let A be an infinite radially symmetric antitree with L = ∞. Then  which completes the proof.
Theorem 9.6. Let A be an infinite radially symmetric antitree with sphere numbers satisfying (9.4). If ℓ * (A) = inf n≥0 ℓ n > 0, Proof. Suppose ℓ * (A) ≥ 2. Then, by Lemma 9.5, for every n ∈ Z ≥0 , we get where M n := µ (n, n + 2) = {s k s k+1 : (n, n + 2) ∩ I k = ∅}. Since ℓ k ≥ 2 for all k ≥ 0 by assumption, µ is either constant on (n, n + 2) or attains precisely two different values. In the first case, the righthand side is equal to zero. In the second, we obviously get the estimate Thus we end up with the following bound which proves the claim by applying Theorem 9.1. It remains to note that the general case ℓ * (A) > 0 can be reduced to the one with ℓ * (A) ≥ 2 by using the standard scaling argument (see also Remark 9.2).
In fact, one can extend the above result to the case when lengths do not admit a strictly positive lower bound. However, in this case one has to modify (9.4) in an appropriate way.
Since µ is given by (3.4), we get Remark 9.8. In fact, the assumptions on lengths that ℓ n ≤ 1 for all n ≥ 0 and ℓ n = o(1) as n → ∞ as well as monotonicity of sphere numbers are superfluous and we need them for simplicity only. Of course, one can considerably weaken them, however, the analysis becomes more involved and cumbersome.
We finish this section with another result based on [14], which also allows to construct antitrees with absolutely continuous spectrum supported on R ≥0 . Theorem 9.9. Let A be an infinite radially symmetric antitree such that vol(A) = ∞ and L µ = ∞. If there are constants a ∈ R and b ∈ R >0 such that where µ is given by (3.4), then σ ac (H) = R ≥0 .
Proof. As in the proof of Theorem 9.1, we know that the operator H is unitarily equivalent to the operator H. By Theorem 3.1 from [14], where M is defined by (6.6). Straightforward calculations finish the proof.
Remark 9.10. For a string operator defined by (5.9), Theorem 9.1 and Theorem 9.9 also imply that the entropy, respectively, some sort of relative entropy of the corresponding spectral measure is finite (see [4] for details). However, the meaning of this fact for the corresponding quantum graph operator H is unclear to us.
(iv) Clearly, (10.2) coincides with condition (i) of Theorem 5.5 and hence it is necessary. Applying the Cauchy-Schwarz inequality, we get the following estimate: n≥0 ℓ n s n s n+1 Therefore, (10.2) implies condition (ii) of Theorem 5.5, which proves the claim.

Remark 10.2.
(i) Both the discreteness and uniform positivity criteria for H β were obtained in [32,Example 8.6]. Notice that these results are a consequence of the positivity of the combinatorial isoperimetric constant in this case (see [32]). Moreover, using the rough estimate (10.6), one would be able to recover the lower bounds (8.9) and (8.10) from [32].
(ii) It is impossible to apply Theorem 9.1 and Theorem 9.9 to A β (this either can be seen from Proposition 10.1(v) or one can prove that both conditions (9.2) and (9.13) are always violated if sphere numbers grow exponentially). (iii) Since the sphere numbers of A β satisfy s n+2 s n = β 2 for all n ≥ 0, we can apply the results of Section 8. Namely, under the additional assumption ℓ * (A β ) > 0, we conclude that the absolutely continuous spectrum of H is in general empty. In particular, it is always the case if ℓ * (A β ) = ∞ (Theorem 8.1). Moreover, assuming that {ℓ n } n≥0 is a finite set, by Theorem 8.3, σ ac (H) = ∅ would imply that the sequence {ℓ n } n≥0 is eventually periodic. (iv) Notice that the isoperimetric constant is given by (see (7.3))

10.2.
Polynomially growing antitrees. Fix q ∈ Z ≥1 and let A q be the antitree with sphere numbers s n = (n + 1) q , n ≥ 0 (the case q = 1 is depicted in Figure 1). Suppose that {ℓ n } n≥0 are the lengths. Notice that vol(A q ) = n≥0 (n + 1) q (n + 2) q ℓ n . (10.7) Then the basic spectral properties of the corresponding quantum graph operator are contained in the following proposition. Assume in addition that (10.8) is satisfied, that is, H q is self-adjoint. (k 2 + 3k + 2) q ℓ k k≥n ℓ k (k 2 + 3k + 2) q = 0.
the claim is immediate from Theorem 9.6.
Remark 10.4. A few remarks are in order.
(i) The antitree A q and the corresponding Kirchhoff Laplacian H have been considered in [32,Example 8.7]. The analysis of spectral properties (in particular, spectral estimates) is a rather delicate task in this case since the combinatorial isoperimetric constant of A q is equal to 0. We were able to describe basic spectral properties of H q only due to the presence of radial symmetry. Spectral properties of Kirchhoff Laplacians without radial symmetry seems to be a rather complicated problem -even the self-adjointness problem (modulo some recent criteria obtained in [17]) is unclear to us at the moment. Let us only mention that there are examples of metric antitrees having finite volume and such that the corresponding Kirchhoff Laplacian has infinite deficiency indices [33]. (ii) It can be demonstrated by examples that the conditions ℓ n = o(n −1 ) (resp., ℓ n = O(n −1 )) as n → ∞ are not necessary for the discreteness (resp., positivity). However, they are in a certain sense sharp (see [32,Lemma 8.9] and also Example 10.6 below). (iii) Since s n+2 = s n (1 + o(1)), we can't apply the results of Section 8 (see Hypothesis 8.1). Moreover, Proposition 10.3(vi) shows that in general H q has absolutely continuous spectrum supported on R ≥0 . However, Theorem 9.1 is a consequence of [4, Theorem 2], which allows a presence of a rather rich singular (continuous) spectrum.
Using the same argument as in the proof of the upper bound in (A.5), we arrive at the desired estimates.
However, seems this does not lead to a closed formula anyway. 5 See, e.g., https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations 6 See, e.g., https://en.wikipedia.org/wiki/Atan2 Since µ ≡ const on (t 1 , L) in the above example, the corresponding operator H can be considered as a coupling of a weighted Sturm-Liouville operator acting on (0, t 1 ) and the free Schrödinger operator −d 2 /dx 2 on (t 1 , L). This explains the result in the above example. Moreover, it also implies that the spectrum of H is purely absolutely continuous and coincides with R ≥0 if L = ∞.
Thus after some calculations we compute Applying Theorem 9.9, we conclude that σ ac (H) = R ≥0 . ♦