Spectral Theory of Infinite Quantum Graphs

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

During the last two decades, quantum graphs became an extremely popular subject because of numerous applications in mathematical physics, chemistry and engineering. Indeed, the literature on quantum graphs is vast and extensive and there is no chance to give even a brief overview of the subject here. We only mention a few recent monographs and collected works with a comprehensive bibliography: [10], [11], [28] and [43]. The notion of quantum graph refers to a graph G considered as a one-dimensional simplicial complex and equipped with a differential operator ("Hamiltonian"). The idea has it roots in the 1930s when it was proposed to model free electrons in organic molecules [78,87]. It was rediscovered in the late 1980s and since that time it found numerous applications. Let us briefly mention some of them: superconductivity theory in granular and artificial materials [6,85], microelectronics and waveguide theory [31,72,73], Anderson localization in disordered wires [1,2,27], chemistry (including studying carbon nanostructures) [7,26,52,62,79], photonic crystal theory [8,34,60], quantum chaotic systems [43,53], and others. These applications of quantum graphs usually involve modeling of waves of various nature propagating in thin branching media which looks like a thin neighborhood Ω of a graph G. A rigorous justification of such a graph approximation is a nontrivial problem. It was first addressed in the situation where the boundary of the "fat graph" is Neumann (see, e.g., [63,86]), a full solution was obtained only recently [16,30]. The Dirichlet case is more difficult and a work remains to be done (see, e.g., a survey by D. Grieser in [28] which contains a nice overview of the subject).
From the mathematical point of view, quantum graphs are interesting because they are a good model to study properties of quantum systems depending on geometry and topology of the configuration space. They exhibit a mixed dimensionality being locally one-dimensional but globally multi-dimensional of many different types. To the best of our knowledge, however, their analysis always includes the assumption that there is a positive lower bound on the lengths of the graph edges. Our main aim is to investigate spectral properties of quantum graphs avoiding this rather restrictive hypothesis on the geometry of the underlying metric graph G.
To proceed further we need to introduce briefly some notions and structures (a detailed description is given in Section 2). Let G d = (V, E) be a discrete graph with finite or countably infinite sets of vertices V = {v k } and edges E = {e j }. For two different vertices u, v ∈ V we shall write v ∼ u if there is an edge e ∈ E connecting v with u. For every v ∈ V, E v denotes the set of edges incident to the vertex v. To simplify our considerations, we assume that the graph G is connected and there are no loops and multiple edges (these assumptions are of technical character and they can be made without loss of generality because one is always achieve that they are satisfied by adding 'dummy' vertices to the graph). In what follows we shall also assume that G d is equipped with a metric, that is, each edge e ∈ E is assigned with the length |e| = l e ∈ (0, ∞) in a suitable way. A discrete graph G d equipped with a metric | · | is called a metric graph and is denoted by G = (G d , | · |). Identifying every edge e with the interval (0, |e|) one can introduce the Hilbert space L 2 (G) = e∈E L 2 (e) and then the Hamiltonian H which acts in this space as the (negative) second derivative − d 2 dx 2 e on every edge e ∈ E. To give H the meaning of a quantum mechanical energy operator, it must be self-adjoint. To make it symmetric, one needs to impose appropriate boundary conditions at the vertices. Kirchhoff conditions (4.1) or, more generally, δ-type conditions with interactions strength α : are the most standard ones (cf. [11]). The first question which naturally appears in this context is, of course, whether the corresponding minimal symmetric operator H α (see Section 3 for a precise definition of H α ) is self-adjoint in L 2 (G). This problem is well understood in the case of finite graphs, that is, when both sets V and E are finite (see, e.g., [58], [11]). To the best of our knowledge, in the case when both sets V and E are countably infinite, the self-adjointness of H α was established under the assumptions that inf e∈E |e| > 0 and the interactions strength α : V → R is bounded from below in a suitable sense (see, e.g., [11, Chapter I] and [64]). The subsequent analysis of H α was then naturally performed only under these rather restrictive assumptions on G and α.
We propose a new approach to investigate spectral properties of infinite quantum graphs. To this goal, we exploit the boundary triplets machinery [22,40,89], a new powerful approach to extension theory of symmetric operators (see Appendix A for further details and references). Consider in L 2 (G) the minimal operator where W 2,2 0 (e) denotes the standard Sobolev space on the edge e ∈ E. Clearly, H min is a closed symmetric operator in L 2 (G) with deficiency indices n ± (H min ) = 2#(E). In particular, the deficiency indices are infinite when G contains infinitely many edges and hence in this case the description of self-adjoint extensions and the study of their spectral properties is a very nontrivial problem. Notice that the boundary triplets approach enables us to parameterize the set of all self-adjoint (respectively, symmetric) extensions of H min in terms of self-adjoint (respectively, symmetric) "boundary linear relations" if one has a suitable boundary triplet for the adjoint operator H * min =: H max . It turns out (see Proposition 3.3) that the boundary relation (to be more precise, its operator part) parameterizing the quantum graph operator H α is unitarily equivalent to the weighted discrete Laplacian h α defined in ℓ 2 (V; m) by the following expression where the weight functions m : V → (0, ∞) and b : V × V → [0, ∞) are given by Therefore, spectral properties of the quantum graph Hamiltonian H α and the discrete Laplacian h α are closely connected. For example, we show that (see Theorem 3.5): (i) The deficiency indices of H α and h α are equal. In particular, H α is selfadjoint if and only if h α is self-adjoint.
Assume additionally that the operator H α (and hence also the operator h α ) is self-adjoint. Then: (ii) H α is lower semibounded if and only if h α is lower semibounded.
(iii) The total multiplicities of negative spectra of H α and h α coincide. In particular, H α is nonnegative if and only if the operator h α is nonnegative. Moreover, negative spectra of H α and h α are discrete simultaneously.
The spectrum of H α is purely discrete if and only if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and the spectrum of h α is purely discrete.
Spectral theory of discrete Laplacians on graphs has a long and venerable history due to its numerous applications in probability (e.g., random walks and Markov processes) and physics (see the monographs [17], [19], [23], [66], [93], [94] and references therein). If inf e∈E |e| = 0, then the corresponding discrete Laplacian h α might be unbounded even if α ≡ 0. A significant progress in the study of unbounded discrete Laplacians has been achieved during the last decade (see the surveys [49], [50]) and we apply these recent results to investigate spectral properties of quantum graphs in the case when inf e∈E |e| = 0. For example, using (i), we establish a Gaffney type theorem (see Theorem 4.9 and Remark 4.10) by simply applying the corresponding result for discrete operators (see [46,Theorem 2]): if G equipped with a natural path metric is complete as a metric space, then H 0 is self-adjoint. Combining (iv) and (v) with the Cheeger type and the volume growth estimates for discrete Laplacians (see [9], [35], [49], [51]), we prove several spectral estimates for H 0 . In particular, we prove necessary (Theorem 4.19(iii)) and sufficient (Theorem 4.18(iii)) discreteness conditions for H 0 . In the case #E = ∞, it follows from (vi) that the condition inf e∈E |e| = 0 is necessary for the spectrum of H 0 to be discrete and this is the very reason why the discreteness problem has not been addressed previously.
Let us also stress that some of our results are new even if inf e∈E |e| > 0. In this case the discrete Laplacian h 0 is bounded and hence we immediately conclude by applying (i) that H α is self-adjoint for any α : V → R (Corollary 5.2). On the other hand, h 0 is bounded if and only if the weighted degree function Deg : V → R defined by [21]). Therefore, H α is self-adjoint for any α : V → R in this case too (Lemma 5.1). Let us stress that the condition inf e∈E |e| > 0 is sufficient for Deg to be bounded on V, however, it is not necessary (see Example 4.7).
The duality between spectral properties of continuous and discrete operators on finite graphs and networks was observed by physicists in the 1960s and then by mathematicians in the 1980s. For a particular class, the so-called equilateral graphs, it is even possible to prove a sort of unitary equivalence between continuous and discrete operators [13,25,76,77] (actually, this can also be viewed as the analog of Floquet theory for periodic Sturm-Liuoville operators, cf. [3]). Furthermore, it is not difficult to discover certain connections just by considering the corresponding quadratic forms. Namely, let f be a continuous compactly supported function on the metric graph G, which is linear on every edge. Setting f V := f ↾ V , we then get (see Remark 3.7 for more details) (1.4) , then the closures of both forms t Hα and t hα are regular Dirichlet forms whenever the corresponding graph G is locally finite (cf. [38]). Even more, every regular Dirichlet form on a discrete graph is of the above form (1.4) (see [38], [51]). This fact, in particular, explains the interest paid to discrete Laplacians on graphs. Clearly, (1.4) establishes a close connection between the corresponding Markovian semigroups as well as between Markov processes on the corresponding graphs. However, let us stress that it was exactly the above statement (iii) which helped us to improve and complete one result of G. Rozenblum and M. Solomyak [83] on the behavior of the heat semigroups generated by H 0 and h 0 (see Theorem 5.17 and Remark 5.18): for D > 2 the following equivalence holds Here C 1 and C 2 are positive constants, which do not depend on t. Let us also mention that the estimates of this type are crucial in proving Rozenblum-Cwikel-Lieb (CLR) type estimates for both H α and h α (see Section 5.2). Our results continue and extend the previous work [54,55,56] and [57] on 1-D Schrödinger operators and matrix Schrödinger operators with point interactions, respectively. Notice that (see Example 3.6) in this case the line or a half-line can be considered as the simplest metric graph (a regular tree with d = 2) and then the corresponding discrete Laplacian is simply a Jacobi (tri-diagonal) matrix (with matrix coefficients in the case of matrix Schrödinger operators).
Let us now finish the introduction by briefly describing the content of the article. The core of the paper is Section 2, where we construct a suitable boundary triplet for the operator H max (Theorem 2.2 and Corollary 2.4) by applying an efficient procedure suggested recently in [55], [69] (see also Appendix A.4). The central result of Section 2 is Theorem 2.8, which describes basic spectral properties (selfadjointness, lower semiboundedness, spectral estimates, etc.) of proper extensions H Θ , H min ⊂ H Θ ⊂ H max , given by (1.5) in terms of the corresponding properties of the boundary relation Θ. In particular, (1.5) establishes a one-to-one correspondence between self-adjoint (respectively, symmetric) linear relations in an auxiliary Hilbert space H and self-adjoint (respectively, symmetric) extensions of the minimal operator H min .
In Section 3 we specify Theorem 2.8 to the case of the Hamiltonian H α . First of all, we find the boundary relation parameterizing the operator H α in the sense of (1.5). As it was already mentioned, its operator part is unitarily equivalent to the discrete weighted Laplacian (1.2)-(1.3) and hence this fact establishes a close connection between spectral properties of H α and h α (Theorem 3.5).
In Sections 4 and 5, we exploit recent advances in spectral theory of discrete weighted Laplacians and prove a number of results on quantum graphs with Kirchhoff and δ-couplings at vertices avoiding the standard restriction inf e∈E |e| > 0. More specifically, the case of Kirchhoff conditions is considered in Section 4, where we prove several self-adjointness results including the Gaffney-type theorem and also provide estimates on the bottom of the spectrum as well as on the essential spectrum of H 0 . We discuss the self-adjointness of H α in Section 5.1. On the one hand, we show that H α is self-adjoint for any α : V → R whenever the weighted degree function Deg is bounded on V. In the case when Deg is locally bounded on V, we prove self-adjointness and lower semiboundedness of H α under certain semiboundedness assumptions on α. We also demonstrate by simple examples that these results are sharp. Section 5.2 is devoted to CLR-type estimates for quantum graphs. In Section 5.3 we investigate spectral types fo H α . Moreover, using the Cheeger-type estimates for h α , we prove several spectral bounds for H α .
As it was already mentioned, Theorem 2.8 is valid for all self-adjoint extensions of H min , however, the corresponding boundary relation may have a complicated structure when we go beyond the δ couplings. In Section 6, we briefly discuss the case of the so-called δ ′ s -couplings. It turns our that the corresponding boundary operator is a difference operator, however, its order depends on the vertex degree function of the underlying discrete graph.
In Appendix A we collect necessary definitions and facts about linear relations in Hilbert spaces, boundary triplets and Weyl functions. H and H denote separable complex Hilbert spaces; I H and O H are, respectively, the identity and the zero maps on H; I n := I C n and O n := O C n . By C(H) and C(H) we denote, respectively, the sets of closed linear operators and relations in H; S p (H) is the two-sided Neumann-Schatten ideal in H, p ∈ (0, ∞]. In particular, S 1 (H), S 2 (H) and S ∞ (H) denote the trace ideal, the Hilbert-Schmidt ideal and the set of compact operators in H.
Let T = T * be a self-adjoint linear operator (relation) in H. For a Borel set Ω ⊆ R, by E Ω (T ) we denote the spectral projection of T ; T − := T E (−∞,0) (T ) and is the total multiplicity of the negative spectrum of T . Note that κ − (T ) is the number (counting multiplicities) of negative eigenvalues of T if the negative spectrum of T is discrete. In this case we denote by λ j (T ) := λ j (|T − |) their absolute values numbered in the decreasing order and counting their multiplicities.

Boundary triplets for graphs
Let us set up the framework. Let G d = (V, E) be a discrete (undirected) graph, that is, V is a finite or countably infinite set of vertices and E is a countably infinite set of edges. For two vertices v, u ∈ V we shall write v ∼ u if there is an edge e u,v ∈ E connecting v with u. For every v ∈ V, we denote the set of edges incident to the vertex v by E v and is called the degree (or combinatorial degree) of a vertex v ∈ V. A path P of length n ∈ N is a subset of vertices {v 0 , v 1 , . . . , v n } ⊂ V such that n vertices {v 0 , v 1 , . . . , v n−1 } are distinct and v k−1 ∼ v k for all k ∈ {1, . . . , n}. A graph G d is called connected if for any two vertices v andṽ there is a path P = {v 0 , v 1 , . . . , v n } connecting u and v, that is, u = v 0 and v = v n . We also need the following assumptions on the geometry of G: Hypothesis 2.1. G d is connected and there are no loops and multiple edges.
Let us assign each edge e ∈ E with length |e| ∈ (0, ∞) 1 and direction 2 , that is, each edge e ∈ E has one initial e o and one terminal vertex e i . In this case G = (V, E, | · |) = (G d , | · |) is called a metric graph. Moreover, every edge e ∈ E can be identified with the interval (0, |e|) and hence we can introduce the Hilbert space Let us equip G with the Laplace operator. For every e ∈ E consider the maximal operator H e,max acting on functions f ∈ W 2,2 (e) as a negative second derivative. Now consider the maximal operator on G defined by For every f e ∈ W 2,2 (e) the following quantities are well defined. We begin with the simple and well known fact (see, e.g., [55]).

5)
is a boundary triplet for H e,max . Moreover, the corresponding Weyl function M 0 e : C → C 2×2 is given by Proof. The proof is straightforward and we leave it to the reader. 1 We shall always assume that there are no edges having an infinite length, however, see Remark

3.1(ii). 2 This means that the graph G d is directed
It is easy to see that the Green's formula holds for all f , g ∈ dom(H max ) ∩ L 2 c (G), where L 2 c (G) is a subspace consisting of functions from L 2 (G) vanishing everywhere on G except finitely many edges, and the asterisk denotes complex conjugation. One would expect that a boundary triplet for H max can be constructed as a direct sum Π = ⊕ e∈E Π 0 e of boundary triplets Π 0 e , however, it is not true once inf e∈E |e| = 0 (see [55] for further details). Using Theorem A.8, we proceed as follows (see also [55,Section 4]). For every e ∈ E we set R e := |e|I 2 , Q e := lim that is, .
(2.10) Clearly, Π e = {C 2 , Γ 0,e , Γ 1,e } is also a boundary triplet for H max,e . In addition, the following claim holds. is a boundary triplet for the operator H max . Moreover, the corresponding Weyl function is given by We shall also need another boundary triplet for H max , which can be obtained from the triplet Π by regrouping all its components with respect to the vertices: and (2.16) Proof. Every f ∈ H andf ∈ H G can be written as follows Clearly, U is a unitary operator and moreover This completes the proof.
Let us also mention other important relations. where M is the Weyl function corresponding to the triplet Π constructed in Theorem 2.2 and U is the operator defined by (2.17).

21)
and Corollary 2.7. Let M G be the Weyl function corresponding to the boundary triplet Proof. It is an immediate consequence of Lemma 2.3 and (2.19).
Let Θ be a linear relation in H G and define the following operator In particular, H Θ is self-adjoint if and only if so is Θ. Assume in addition that Θ is a self-adjoint linear relation (hence H Θ is also self-adjoint). Then: (iv) H Θ is lower semibounded if and only if the same is true for Θ.
(v) H Θ is nonnegative (positive definite) if and only if Θ is nonnegative (positive definite). (vi) The total multiplicities of negative spectra of H Θ and Θ coincide, (2.24) (vii) For every p ∈ (0, ∞] the following equivalence holds If the negative spectrum of H Θ (or equivalently Θ) is discrete, then for every γ ∈ (0, ∞) the following equivalence holds . Then for every p ∈ (0, ∞] the following equivalence holds for the corresponding Neumann-Schatten ideals The spectrum of H Θ is purely discrete if and only if #{e ∈ E : |e| > ε} is finite for every ε > 0 and the spectrum of the linear relation Θ is purely discrete. Proof. Items (i), (ii), (iii) and (x) follow from Theorem A.3. Item (iv) follows from Theorem A.6 and Corollary 2.7. Consider the boundary triplet Π constructed in Theorem 2.2 and the corresponding Weyl function M given by (2.12). Clearly, Noting that H 0 e := H e,max ↾ ker(Γ 0,e ) = H F e is the Friedrichs extension of H e,min = (H e,max ) * , we immediately conclude that Now items (v)-(viii) follow from Theorem A.5 and item (ix) follows from Theorem A.7.
Finally, it follows from (2.29) and (2.30) that the spectrum of H F is purely discrete if and only if #{e ∈ E : |e| > ε} is finite for every ε > 0. This fact together with Theorem A.3(iv) implies item (xi).
Remark 2.9. The analogs of statements (iii) and (iv) of Theorem 2.8 were obtained in [64] under the additional very restrictive assumption inf e∈E |e| > 0. Notice that if the latter holds, then the regularization (2.9) is not needed and one can construct a boundary triplet for the maximal operator H max by summing up the triplets (2.5).

Parameterization of quantum graphs with δ-couplings
Turning to a more specific problem, we need to make further assumptions on the geometry of a connected metric graph G. Let α : V → R be given and equip every vertex v ∈ V with the so-called δ-type vertex condition:

2)
Let us define the operator H α as a closure of the operator H 0 α given by Remark 3.1. A few remarks are in order: that a Kirchhoff-type boundary condition at v 0 (as well as (3.2)) leads to an operator which is not closed. Moreover, it turns out that its closure gives rise to Dirichlet boundary condition at v 0 , i.e., disconnected edges. (ii) Assumption (3.1) is of a technical character. Of course, the case of edges having an infinite length would require separate considerations in Section 2 and this will be done elsewhere. On the other hand, the case when all edges have finite length but there is no uniform upper bound can be reduced to the case of graphs satisfying (3.1) either by adding additional "dummy" vertices or by slight modifications in the considerations of Section 2. Note also that those allow to include situations when the graph has loops and multiple edges (cf. Hypothesis 2.1).
Let us emphasize that the operator H α is symmetric. Moreover, simple examples show that H α might not be self-adjoint.

Example 3.2 (1-D Schrödinger operator with δ-interactions).
Consider the positive semi-axis R + and let {x k } k∈Z+ ⊂ [0, ∞) be a strictly increasing sequence such that x 0 = 0 and x k ↑ +∞. Considering x k as vertices and the intervals e k = (x k−1 , x k ) as edges, we end up with the simplest infinite metric graph. Notice that for every real sequence α = {α k } k∈Z+ with α 0 = 0 conditions (3.2) take the following form: f ′ (0) = 0 and The operator H α is known as the one-dimensional Schrödinger operator with δinteractions on X = {x k } k∈N (see, e.g., [4]), and the corresponding differential expression is given by It was proved in [55] that H X,α is self-adjoint if k |e k | 2 = ∞ (the latter is known in the literature as the Ismagilov condition, see [47]). On the other hand (see [55,Proposition 5.9]), if k |e k | 2 < ∞ and in addition |e k−1 | · |e k+1 | ≥ |e k | 2 for all k ∈ N, then the operator H X,α is symmetric with n ± (H X,α ) = 1 whenever α = {α k } k∈N satisfies the following condition This effect was discovered by C. Shubin Christ and G. Stolz [91, pp. 495-496] in the special case |e k | = 1/k and α k = −(2k + 1), k ∈ N. For further details and results we refer to [56], [71]. ♦ Our main aim is to find a boundary relation Θ α parameterizing the operator H α in terms of the boundary triplet Π G given by (2.13)-(2.15). First of all, notice that at each vertex v ∈ V the boundary conditions (3.2) have the following form (2.21) and (2.22)) and the (3.7) It is easy to check that these matrices satisfy the Rofe-Beketov conditions (see Proposition A.1), that is Here Γ 0 0 and Γ 0 1 are given by (2.21) and (2.22), respectively. In view of (2.20), we get and R = ⊕ e∈E R e , Q = ⊕ e∈E Q e and U are defined by (2.8) and (2.17), respectively.
Thus we are led to specification of the boundary relation parameterizing the operator H 0 α . Namely, consider now the linear relation Θ 0 where H G,c consists of vectors of H G having only finitely many nonzero coordinates. It is not difficult to see that Θ 0 α is symmetric and hence it admits the decomposition (see Appendix A.1) where Noting that Finally, take g ∈ H G,c and consider . Therefore, define g v ∈ H op by where the function m : Clearly, and hence f v ∈ dom(Θ 0 α ). Moreover, (3.10) immediately implies that More precisely, we define the operator h 0 . Moreover, h 0 α is symmetric and let us denote its closure by h α . Thus we proved the following result.
where Θ α is a linear relation in H G defined as the closure of Θ 0 α given by (3.9). Moreover, the operator part Θ op α of Θ α is unitarily equivalent to the operator h α = h 0 α acting in ℓ 2 (V). We also need another discrete Laplacian. Specifically, in the weighted Hilbert space ℓ 2 (V; m) we consider the minimal operator defined by the following difference expression is unitarily equivalent to the operator h 0 α defined by (3.13), (3.15) and acting in ℓ 2 (V). Proof. It suffices to note thath In the following we shall use h α as the symbol denoting the closures of both operators. Now we are ready to formulate our main result.
Theorem 3.5. Assume that Hypotheses 2.1 and 3.1 are satisfied. Let α : V → R and H α be a closed symmetric operator associated with the graph G and equipped with the δ-type coupling conditions (3.2) at the vertices. Let also h α be the discrete Laplacian defined either by (3.13) in ℓ 2 (V) or by (3.17) in ℓ 2 (V; m), where the functions m : V → (0, ∞) and b : V × V → [0, ∞) are given by (3.11) and (3.14), respectively. Then: (i) The deficiency indices of H α and h α are equal and In particular, H α is self-adjoint if and only if h α is self-adjoint. Assume in addition that H α (and hence also h α ) is self-adjoint. Then: (ii) The operator H α is lower semibounded if and only if the operator h α is lower semibounded. (iii) The operator H α is nonnegative (positive definite) if and only if the operator h α is nonnegative (respectively, positive definite). (iv) The total multiplicities of negative spectra of H α and h α coincide, (v) Moreover, the following equivalence holds for all p ∈ (0, ∞]. In particular, negative spectra of H α and h α are discrete simultaneously. , then the following equivalence holds for all γ ∈ (0, ∞) The spectrum of H α is purely discrete if and only if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and the spectrum of the operator h α is purely discrete.
Proof. We only need to comment on the first equality in (3.18) since the rest immediately follows from Theorem 2.8 and Proposition 3.3. However, the first equality in (3.18) follows from the equality of deficiency indices of the operator h α . Indeed, n + (h α ) = n − (h α ) by the von Neumann theorem since h α commutes with the complex conjugation.
Let us demonstrate Theorem 3.5 by applying it to the 1-D Schrödinger operator with δ-interactions (3.5) considered in Example 3.2.
Example 3.6. Let H X,α be the Schrödinger operator (3.5) with δ-interactions on the semi-axis R + . Recall that in this case V = {x k } k∈Z+ and E = {e k } k∈N , where e k = (x k−1 , x k ). By (3.11) and (3.14), we get , we see that the difference expression (3.13) is just a three-term requrence relation for all k ∈ N. Hence the corresponding operator h α is the minimal operator associated in ℓ 2 (Z + ) with the Jacobi (tri-diagonal) matrix (3.23) In this particular case Theorem 3.5 was established in [55] and in the recent paper [57] it was extended to the case of Schrödinger operators in a space of vector-valued functions. ♦ defines an equivalent norm on L cont . On the other hand, for every f ∈ L cont we get However, one can easily check that the latter is the quadratic form of the discrete operator h α defined in ℓ 2 (V; m) by (3.17), that is, the following equality holds for every f ∈ ℓ 2 c (V; m).

Quantum graphs with Kirchhoff vertex conditions
As in Section 3, if it is not explicitly stated, we shall always assume that G satisfies Hypotheses 2.1 and 3.1. In this section we restrict ourselves to the case α ≡ 0, that is, we consider the quantum graph with Kirchhoff vertex conditions    f is continuous at v,    (iv) The spectrum of H 0 is purely discrete if and only if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and the spectrum of the operator h 0 is purely discrete.
Our next goal is to use the spectral theory of discrete Laplacians (4.2) to prove new results for quantum graphs.
4.1. Intrinsic metrics on graphs. During the last decades a lot of attention has been paid to the study of spectral properties of the discrete Laplacian (4.2). Let us recall several basic concepts. Suppose that the metric graph G = (V, E, | · |) satisfies Hypotheses 2.1 and 3.1. The function Deg : V → (0, ∞) defined by = 0 for all v ∈ V and satisfies the triangle inequality. Notice that every function p : E → (0, ∞) generates a path pseudo metric ̺ p on V with respect to the graph G via Here the infimum is taken over all paths connecting u and v. Following [36] (see also [9,49]), a pseudo metric ̺ on V is called intrinsic with respect to the graph G if holds on V. Notice that for any given locally finite graph an intrinsic metric always exists.
(a) Let p : E → (0, ∞) be defined by It is straightforward to check that the corresponding path pseudo metric ̺ p is intrinsic (see [46, Example 2.1], [49]). (b) Another pseudo metric was suggested in [18]. Namely, let ̺ be a path pseudo metric generated by the function p : E → (0, ∞) It was shown in [46] that this metric is equivalent to the metric  generates an intrinsic (with respect to the graph G) path metric ̺ 0 on V. Moreover, ̺ 0 is the maximal intrinsic path metric on V, that is, for any intrinsic path metric ̺ on V the following inequality holds for all u, v ∈ V Proof. First of all, notice that for the functions (4.3) the condition (4.7) takes the following form for every v ∈ V. Clearly (4.11) holds with ̺ = ̺ 0 for all v ∈ V with equality instead of inequality since Moreover, the latter also implies that ̺(u, v) ≤ ̺ 0 (u, v) for any u ∼ v and any intrinsic metric ̺. Since ̺ is also a path metric on V, this inequality together with (4.6) immediately completes the proof.
For any v ∈ V and r ≥ 0, the distance ball B r (v) with respect to a pseudo metric ̺ is defined by (4.12) Finally for a set X ⊂ V, the combinatorial neighborhood of X is given by Ω(X) := {u ∈ V : u ∈ X or there exists v ∈ X such that u ∼ v}.
(4.13) 4.2. Self-adjointness of H 0 . In this and the following subsections we shall always assume that the metric graph G satisfies Hypotheses 2.1 and 3.1. We begin with the following result. then the operator H 0 is self-adjoint.
Proof. Consider the corresponding boundary operator h 0 defined by (4.2). Since Deg is bounded on V, the operator h 0 is bounded on ℓ 2 (V; m) (see (4.5)) and hence self-adjoint. It remains to apply Corollary 4.1(i).
As an immediate corollary of this result we obtain the following widely known sufficient condition (cf. [11,Theorem 1.4.19]). Proof. By Theorem 4.4, it suffices to check that Deg is bounded on V: A few remarks are in order: Remark 4.6.
(i) Numerous graphs considered both in theoretical purposes and in applications belong to this category [11]. Prominent examples are equilateral graphs (see, e.g., [76,77]) and periodic graphs (with a finite number of edges in the period cell).  Continuing this procedure to infinity we end up with an infinite metric graph (note that this type of graphs is called rooted trees) such that inf e∈E |e| = 0 but sup e∈E |e| = 1. It is easy to see that Hence, by Lemma 5.1, the corresponding Hamiltonian H α is self-adjoint for any α : V → R. ♦ The next result shows that we can replace uniform boundedness of the weighted degree function by the local one (in a suitable sense of course).
Theorem 4.8. Let ̺ be an intrinsic pseudo metric on V such that the weighted degree Deg is bounded on every distance ball in V. Then H 0 is self-adjoint.
As an immediate corollary we arrive at the following Gaffney type theorem for quantum graphs. Proof. By Hypothesis 3.1, the discrete graph G d = (V, E) is locally finite. Hence by a Hopf-Rinow type theorem [46], (V, ̺ 0 ) is complete as a metric space if and only if the distance balls in (V, ̺ 0 ) are finite. The latter immediately implies that the weighted degree Deg is bounded on every distance ball in (V, ̺ 0 ). It remains to apply Theorem 4.8.
Remark 4.10. Notice that Theorem 4.9 can be seen as the analog of the classical result of Gaffney [39] (see also [41,Chapter 11] for further details), who established self-adjointness of the Dirichlet Laplacian on a complete Riemannian manifold. Indeed, | · | generates a natural path metric on a metric graph G = (V, E, | · |) and it is easy to check that G equipped with this metric is complete as a metric space if and only if (V, ̺ 0 ) is complete as a metric space.
On the one hand, simple examples demonstrate that Theorem 4.9 is sharp. Indeed, consider the second derivative on an interval (0, ℓ) with ℓ ∈ (0, ∞]. As in Example 3.2, let {x k } k∈Z+ be a strictly increasing sequence such that x k ↑ ℓ as k → ∞. In this case Kirchhoff conditions are equivalent to the continuity of a function and its derivative at every vertex x k (see (3.4)). The corresponding operator is self-adjoint only if ℓ = ∞. However, on the other hand, we can improve Theorem 4.9 by replacing the natural path metric ̺ 0 by another path metric (which is not intrinsic!) generated by the weight function m. is infinite. By Theorem 6 from [51], the latter implies that the operator h 0 is self-adjoint in ℓ 2 (V; m). It remains to apply Corollary 4.1(i). then the operator H 0 is self-adjont.
Proof. Clearly, every infinite geodesic in (V, ̺ m ) has infinite length if (4.16) is satisfied. According to Hypothesis 3.1, G is a locally finite graph and hence combining the Hopf-Rinow type theorem [46] with Proposition 4.11 we finish the proof. Example 4.14. Let G ⊂ R 2 be a planar graph constructed as follows (see the figure depicted below). Let X = {x k } k≥1 ⊂ [0, ∞) be a strictly increasing sequence with x 1 = 0. We set V = X × {−1, 0, 1} and denote v k,n = (x k , n), k ∈ N and n ∈ {−1, 0, 1}. Now we define the set of edges by the following rule: v n,k ∼ v m,j if either n = m and |k − j| = 1 or k = j = 0 and |n − m| = 1. Finally, we assign lengths as the usual Euclidian length in R 2 : the length of every vertical edge is equal to 1, and the length of the horizontal edge e v k,0 ,v k+1,0 is equal to that is, the points x k accumulate at infinity. On the other hand, for all v ∈ V = X × {−1, 0, 1}, and hence it is not difficult to see that (V, ̺ m ) is always complete. Therefore, the corresponding operator H 0 is always self-adjoint in view of Corollary 4.12 ♦ Remark 4.15. The graphs considered in Examples 4.7 and 4.14 belong to a special class of graphs, the so-called trees. More precisely, a path P = {v 0 , v 1 , . . . , v n } ⊂ V is called a cycle if v 0 = v n . A connected graph G = (V, E) without cycles is called a tree. Notice that for any two vertices u, v on a tree T = (V, E) there is exactly one path P connecting u and v and hence every path on a tree is a geodesic with respect to a path metric.
Let us finish this subsection with some sufficient conditions for H 0 to have nontrivial deficiency indices. Let ̺ 1/2 be a path metric on V generated by the function p 1/2 : E → (0, ∞) defined by (4.17) If (V, ̺ 1/2 ) is not complete as a metric space, we then denote the metric completion of (V, ̺ 1/2 ) by V and V ∞ := V \ V. By [18,Lemma 2.1], every function f : V → R such that the corresponding quadratic form is finite, is uniformly Lipschitz with respect to the metric ̺ 1/2 and hence admits a continuation F to V as a Lipschitz function. Following [18], we set f ∞ := F ↾ V ∞ .  Theorem 4.18. Suppose that the operator H 0 is self-adjoint. Then: The spectrum of H 0 is purely discrete if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and C ess (V) = ∞.
Proof. Let ̺ 0 be a natural path metric on V (see Lemma 4.3). Noting that ̺ 0 is an intrinsic metric on V, let us apply the Cheeger estimates from [9] for the discrete Laplacian h 0 given by (4.2), (4.3). First of all (see [9, Section 2.3]), observe that the weighted area with respect to ̺ 0 is given by Hence in this case the Cheeger estimate for discrete Laplacians (see Theorems 3.1 and 3.3 in [9]) implies the following estimates Combining these estimates with Corollary 4.1(ii)-(iii), we prove (i) and (ii), respectively.
Applying [9,Theorem 3.3] once again, we see that the spectrum of h 0 is purely discrete if C ess (V) = ∞. Corollary 4.1(iv) finishes the proof of (iii).
Let B r (u) be a distance ball with respect to the natural path metric ̺ 0 . Following [45] (see also [49]), we define for a fixed v ∈ V, and Theorem 4.19. Let (V, ̺ 0 ) be complete as a metric space. Then: Proof. By Corollary 4.9, the operator H 0 is self-adjoint. The proof follows from the growth volume estimates on the spectrum of h 0 . More precisely, the following bounds were established in [45] (see also [35,49]): It remains to apply Corollary 4.1(ii)-(iv).
We finish this section with a remark.
Remark 4.20. Connections between inf σ(H 0 ) and inf σ(h 0 ) and also between inf σ ess (H 0 ) and inf σ ess (h 0 ) by means of Theorem A.5 and Theorem A.7 are rather complicated since they involve the corresponding Weyl function, which in our case has the form (2.19). In particular, it would be a rather complicated task to use these connections and then apply the Cheeger-type bounds for h 0 to estimate inf σ(H 0 ) and inf σ ess (H 0 ). For example, the following upper estimate, which easily follows from (2.29), inf σ(H 0 ) ≤ inf σ(H F ) = π 2 sup e∈E |e| 2 seems to be unrelated to inf σ(h 0 ). Surprisingly enough, we have been unaware of Cheeger-type bounds for quantum graphs and this will be done elsewhere.

Spectral properties of quantum graphs with δ-couplings
In this section we are going to investigate spectral properties of the Hamiltonian H α with δ-couplings (3.2) at the vertices. Namely, let α : V → R and the operator H α be defined in L 2 (G) as the closure of (3.3). By Theorem 3.5, its spectral properties correlate with the corresponding properties of the discrete operator h α defined in ℓ 2 (V; m) by (3.17). In this section we shall always assume Hypotheses 2.1 and 3.1. 5.1. Self-adjointness and lower semiboundedness. We begin with the study of the self-adjointness of the operator H α . Our first result can be seen as a straightforward extension of Theorem 4.4.
Lemma 5.1. If the weighted degree function Deg defined by (4.4) is bounded on V, that is, (4.14) is satisfied, then the operator H α is self-adjoint for any α : V → R. Moreover, in this case the operator H α is bounded from below if and only if is clearly self-adjoint. If Deg is bounded on V, then the operator h 0 is bounded and self-adjoint in ℓ 2 (V; m) (see (4.5)). It remains to note that h α = h 0 + A and hence h α is a self-adjoint operator since the self-adjointness is stable under bounded perturbations. Moreover, h α is bounded from below if and only if so is A. The latter is clearly equivalent to (5.1). Theorem 3.5(i)-(ii) completes the proof.
As an immediate corollary we arrive at the following result. Proof. As in the proof of Corollary 4.5, we get It remains to apply Lemma 5.1.

Remark 5.3. A few remarks are in order.
(i) Using the form approach, the self-adjointness claim in Corollary 5.2 was proved in [11,Section I.4.5] under the additional assumption that α deg : V → R is bounded from below, If 0 < inf e∈E |e| ≤ sup e∈E |e| < ∞, then it is easy to see that (5.3) is equivalent to (5.1). (ii) Let us also mention that the graphs constructed in Examples 4.7 and 4.14 do not satisfy the condition of Corollary 5.2, however, they satisfy (4.14) and hence, by Lemma 5.1, the corresponding Hamiltonian H α is self-adjoint for any α : V → R.
The next result allows us to replace the boundedness assumption on the weighted degree by the local boundedness, however, now we need to assume some semiboundedness on α. We begin with the following result.
Proposition 5.4. If the operator H 0 with Kirchhoff vertex conditions is self-adjoint in L 2 (G), then the operator H α with δ-couplings on V is self-adjoint whenever the function α : V → R satisfies (5.1).
Proof. By Corollary 4.1(i), the discrete Laplacian h 0 given by (4.2), (4.3) is a nonnegative self-adjoint operator in ℓ 2 (V; m). On the other hand, (5.1) implies that the multiplication operator A defined by (5.2) is a self-adjoint lower semibounded operator in ℓ 2 (V; m). Noting that ℓ 2 c (V; m) is a core for both h 0 and A since the graph is locally finite, we conclude that the operator h α defined as a closure of the sum of h 0 and A is a lowersemibounded self-adjoint operator in ℓ 2 (V; m) (see [48, Chapter VI.1.6]). It remains to apply Theorem 3.5(i).
Remark 5.5. It follows from the proof of Proposition 5.4 and Theorem 3.5(ii) that the operator H α is lower semibounded in this case.
Combining Proposition 5.4 with the self-adjointness results from Section 4.2, we can extend Corollary 5.2 to a much wider setting. Let us present only one result in this direction.
Corollary 5.6. Let ̺ m be the path metric (4.15), (4.6) on V. If (V, ̺ m ) is complete as a metric space and α : V → R satisfies (5.1), then H α is a lower semibounded self-adjoint operator.
In particular, if the weight function m satisfies (4.16) and inf v∈V α(v) > −∞, then H α is a lower semibounded self-adjoint operator.
Remark 5.7. Let us stress that both conditions (completeness of (V, ̺ m ) and (5.1)) are important. Indeed, 1-D Schrödinger operators with δ-type interactions (see Example 3.2) immediately provide counterexamples. First of all, in this setting completeness of (V, ̺ m ) means that we consider a Schrödinger operator on an unbounded interval (either on the whole line R or on a semi-axis). Clearly, in the case of a compact interval the minimal operator is not self-adjoint even in the case of trivial couplings α ≡ 0. On the other hand, it was proved in [5] that in the case when all δ-interactions are attractive (α k < 0 for all k ∈ N), the operator H α given by In the case inf k∈N (x k+1 − x k ) > 0 the latter is equivalent to inf k∈N α k > −∞.

5.2.
Negative spectrum: CLR-type estimates. Let α : V → [0, ∞) be a nonnegative function on V. The main focus of this section is to obtain the estimates on the number of negative eigenvalues κ − (H −α ) of the operator H −α in terms of the interactions α : V → [0, ∞). Note that by Theorem 3.5(iv), where h −α is the (self-adjoint) discrete Laplacian defined either by (3.13) in ℓ 2 (V) or by (3.17) in ℓ 2 (V; m). Suppose that the discrete Laplacian h 0 defined by (3.17) with α ≡ 0 is a selfadjoint operator in ℓ 2 (V; m) (see Section 4.2). It is well known (cf., e.g., [38]) that in this case h 0 generates a symmetric Markovian semigroup e −th0 (one can easily check that the Beurling-Deny conditions [20,38] are satisfied). Let us consider the corresponding quadratic form in ℓ 2 (V; m): which is a regular Dirichlet form since G is locally finite (see [38,51]). Recall that the functions m and b are given by (  (i) There are constants D > 2 and K > 0 such that There are constants C > 0 and D > 2 such that for all α : is bounded from below and closed and, moreover, the negative spectrum of h −α is discrete and the following estimate holds Remark 5.9.
(i) The constants K and C in Theorem 5.8 are connected by K D ≤ C ≤ e D−1 K D (see [37]). (ii) Since ℓ 2 c (V; m) is a core for both h 0 and A whenever h 0 is essentially selfadjoint, it follows from Theorem 5.8 that the operator h −α is bounded from below and self-adjoint for all α ∈ ℓ D/2 (V; m 1−D/2 ) if (5.7) is satisfied.
Combining Theorem 3.5(iv) with Theorem 5.8, we immediately arrive at the following CLR-type estimate for quantum graphs with δ-couplings at vertices. Theorem 5.10. Assume that h 0 is a self-adjoint operator in ℓ 2 (V; m). Then the following conditions are equivalent: (i) There are constants D > 2 and K > 0 such that (5.7) holds for all f ∈ dom(t 0 ) with q = 2D D−2 . (ii) There are constants C > 0 and D > 2 such that for all α : V → [0, ∞) belonging to ℓ D/2 (V; m 1−D/2 ) the operator H −α is self-adjoint, bounded from below, its negative spectrum is discrete and the following estimate holds The constants K and C are connected by K D ≤ C ≤ e D−1 K D .
Of course, the most difficult part is to check the validity of the Sobolev-type inequality (5.7). However, there are several particular cases of interest when (5.7) is known to be true (see [42], [88], [93] and references therein).
Corollary 5.11. Let the metric graph G = (V, E, |·|) be such that the discrete graph with some constant C(G), which depends only on G.
Proof. By Theorem 5.10, we only need to show that (5.7) holds true. The argument is similar to [65,Theorem 3.7]. Indeed, by [93, Theorem VI.5.2], since G d is a group of polynomial growth, there is a C > 0 such that for all f ∈ ℓ 2 c (V). Combining this inequality with (5.11) and noting that we get (5.7).
Remark 5.12. Notice that in Corollary 5.11 we did not make any additional assumptions on the weight function m. Namely, we only assumed that the edges lengths satisfy (3.1).
In particular, in the case G d = Z D we get the following estimate.
with some constant C N , which depends only on N and m.
It turns out that the later holds in a wider setting and hence we arrive at the following result.
Here the constant C depends only on q, D and V. Proof. By Theorem 3.5(iv), we only need to show that The validity of (5.15) was established in [83,Theorem 3.1] under the additional assumptions inf e∈E |e| > 0 and sup v∈V deg(v) < ∞. In fact, this proof (see also [82, §3]) can be extended line by line to the case of graphs G satisfying (4.14).
In particular, the exponents d and D determined by  14) holds, that is, if h 0 is a bounded operator and, moreover, . It is precisely this fact which allows to prove Proposition 5.14. Note that d = D = N for Schrödinger operators on R N and hence the estimates of the type (5.14) have no analogues in this case.
Equality (5.5) together with Remark 5.16 indicate that there is a close connection between the heat semigroups e −th0 and e −tH0 . In fact, the following holds true.
Proof. By Varopoulos's theorem (see [93,Theorem II.5.2]), (i) and (ii) are equivalent to the validity of the corresponding Sobolev type inequalities. Namely, (i) is equivalent to (5.7) and (ii) is equivalent to the inequality where H 1 (G) is the Sobolev space on G, which coincides with the form domain of the operator H 0 , and q = 2D D−2 and D > 2. Hence it suffices to show that (5.7) is equivalent to (5.19).
First observe that every f ∈ H 1 (G) admits a unique decomposition f = f lin + f 0 , where f lin ∈ H 1 (G) is piecewise linear on G and f 0 ∈ H 1 (G) takes zero values at the vertices V. It is easy to check that Moreover, we have (see Remark 3.7): Next it is easy to see that (5.19) holds for all f = f 0 ∈ H 1 (G) with q > 2 and with a constant C(G) which depends only on sup e∈E |e| and q > 2. Noting that every piecewise linear function we conclude that (i) implies (ii). Clearly, to prove that (ii) implies (i) it suffices to show that every linear function f on a finite interval (a, b) satisfies the estimate where C > 0 is a positive constant which depends only on q > 2. Indeed, we have Applying the Hölder inequality to the left-hand side in (5.21), one gets On the other hand, applying the Cauchy-Schwarz inequality to the right-hand side in (5.21), we arrive at where c(q) > 0 depends only on q > 2. Combining this estimate with (5.21) and (5.22), we obtain (5.20), which implies that holds for all f = f lin ∈ H 1 (G) ∩ L.  ( Here α ∈ c 0 (V) means that the set {v ∈ V : |α(v)| > ε} is finite for every ε > 0.

Proof. It suffices to note that
m ∈ c 0 (V) and then, by the Weyl theorem and Theorem 3.5(viii), we prove the first claim.
The presence of an absolutely continuous spectrum for quantum graphs H 0 with Kirchhoff vertex conditions at vertices is a challenging open problem. To the best of our knowledge, radial trees and classes of graphs that originate form groups (e.g., Cayley graphs) are the only cases where the structure of the continuous spectrum is rather well understood (see, e.g., [12], [29], [33], [92]). In particular, it is shown in [33,Theorem 5.1] that in the case when G is a rooted radial tree with a finite complexity of the geometry, the absolutely continuous spectrum of H 0 is nonempty if and only if G is eventually periodic.
Our next result provides a sufficient condition for H α to have purely singular spectrum.
Theorem 5.20. Assume that inf e∈E |e| > 0 and sup e∈E |e| < ∞. If α : V → R is such that for any infinite path P ⊂ G without cycles Proof. The proof is based on the standard trace class argument [90]. By Corollary 5.2, the operator H α is self-adjoint. Since (5.23) holds for every infinite path P ⊂ G, and the graph G is a countable union of finite subgraphs G k , k ∈ N such that the boundary ∂G k of every subgraph G k is contained in V. Define a new functioñ that is, at every vertex v ∈ V \ V the corresponding boundary condition for Hα is given by (3.2) and at every vertex v ∈ V it has the Dirichlet boundary condition. Let us show that 26) It is easy to see that under the assumptions inf e∈E |e| > 0 and sup e∈E |e| < ∞ the triplet Π = {H G , Γ 0 0 , Γ 0 1 } given by (2.21), (2.22) is a boundary triplet for H max . Next we set where C v,α and D v are given by (3.7), and where (5.29) Observe that the corresponding boundary relations Θ α and Θα parameterizing H α and Hα via the boundary triplet which is finite according to (5.24). Therefore, by Theorem A.3(iv), (5.26) holds true. It remains to note that Hα is the orthogonal sum of operators having discrete spectra and hence the spectrum of Hα is pure point. The Birman-Krein theorem then yields σ ac (H α ) = σ ac (Hα) = ∅.
Corollary 5.21. Let G be a rooted radial tree such that inf e∈E |e| > 0 and sup e∈E |e| < ∞. Let also α : V → R be radial, that is,  [50], [9]. For every subgraph V ⊆ V one defines the modified isoperimetric constant where Area(∂X) := and Moreover, we need the isoperimetric constant at infinity Theorem 5.23. Suppose that the operator H α is self-adjoint. Then: The spectrum of H α is discrete if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and C ess,α (V) = ∞.
Proof. The proof is analogous to that of Theorem 4.18 and we only need to use the corresponding modifications of Cheeger type bounds for the discrete operator h α from [9].

Other boundary conditions
In the present paper our main focus was on the Kirchhoff and δ-type couplings at vertices (see (3.2)). There are several other physically relevant classes of couplings (see, e.g., [11,15,24]). Our main result, Theorem 2.8, covers all possible cases, however, the key problem is to calculate the boundary operator and then to investigate its spectral properties. It turned out that for δ-couplings the corresponding boundary operator is given by the discrete Laplacian (3.17), which attracted an enormous attention during the last three decades. However, for other boundary conditions new nontrivial discrete operators of higher order may arise. For example, this happens in the case of the so-called δ ′ s -couplings. Namely (see [15]), let β : V → R and consider the following boundary conditions at the vertices v ∈ V:    df dxe (v) does not depend on e at the vertex v, Define the corresponding operator H β as the closure of the operator H 0 β given by To avoid lengthy and cumbersome calculations of the corresponding boundary relation Θ β parameterizing H β with the help of the boundary triplet Π constructed in Corollary 2.4, let us consider the kernel L = ker(H max ) of H max as in Remark 3.7. Recall that L = ker(H max ) consists of piecewise linear functions on G and every f ∈ L can be identified with its values on V, {f (e i ), f (e o )} e∈E . Moreover, the L 2 norm of f ∈ L is equivalent to It is not difficult to see that (see also [11, p.27]) Therefore, for every f ∈ L ∩ L 2 c (G) we get Clearly, the right-hand side in (6.3) is a form sum of two difference operators, where the first one is the standard discrete Laplacian, however, the second one gives rise to a difference expression of higher order. In particular, its order at every vertex equals the degree deg(v) of the corresponding vertex v ∈ V. Unfortunately, we are not aware of the literature where the difference operators of this type have been studied. Then Θ C,D is self-adjoint if and only if the following conditions hold: If dim H = N < ∞, then (ii) is equivalent to rank(C|D) = N .
Further details and facts about linear relations in Hilbert spaces can be found in [89,Chapter 14].
A.2. Boundary triplets and proper extensions. Let A be a densely defined closed symmetric operator in a separable Hilbert space H with equal deficiency indices n ± (A) = dim N ±i ≤ ∞, N z := ker(A * − z). is surjective.
A boundary triplet for A * exists since the deficiency indices of A are assumed to be equal. Moreover, n ± (A) = dim(H) and A = A * ↾ ker(Γ). Note also that the boundary triplet for A * is not unique.
An extensionÃ of A is called proper if dom(A) ⊂ dom(Ã) ⊂ dom(A * ). The set of all proper extensions is denoted by Ext(A). (iii) A Θ is symmetric if and only if Θ is symmetric and n ± (A Θ ) = n ± (Θ) holds.
In particular, A Θ is self-adjoint if and only if Θ is self-adjoint. (iv) If A Θ = A * Θ and A Θ = A * Θ , then for every p ∈ (0, ∞] the following equivalence holds is called the Weyl function corresponding to the boundary triplet Π. The Weyl function is well defined and holomorphic on ρ(A 0 ). Moreover, it is a Herglotz-Nevanlinna function (see [22]).
Assume now that A ∈ C(H) is a lower semibounded operator, i.e., A ≥ a I H with some a ∈ R. Let a 0 be the largest lower bound for A, The Friedrichs extension of A is denoted by A F . If Π = {H, Γ 0 , Γ 1 } is a boundary triplet for A * such that A 0 = A F , then the corresponding Weyl function M is holomorphic on C \ [a 0 , ∞). Moreover, M is strictly increasing on (−∞, a 0 ) (that is, for all x, y ∈ (−∞, a 0 ), M (x) − M (y) is positive definite whenever x > y) and the following strong resolvent limit exists (see [22]) We complete this subsection with the following important statement. The implication (ii) ⇒ (i) always holds true (cf. Theorem A.5(i)), however, the validity of the converse implication requires that M tends uniformly to −∞. Let us mention in this connection that the weak convergence of M (x) to −∞, i.e., the relation lim holds or all h ∈ H \ {0} whenever A 0 = A F . Moreover, this relation characterizes Weyl functions of the Friedrichs extension A F among all non-negative (and even lower semibounded) self-adjoint extensions of A (see [59], [22,Proposition 4]). The next new result establishes a connection between the essential spectra of A Θ and Θ and also it can be seen as an improvement of Theorem A.5 (iv).
Theorem A.7. Let A ≥ a 0 I H > 0 and let Π = {H, Γ 0 , Γ 1 } be a boundary triplet for A * such that A 0 = A F . Let also M be the corresponding Weyl function and let Θ = Θ * ∈ C(H) be such that A Θ = A * Θ is lower semibounded. Then the following equivalences hold: Based on these criteria, different regularizations Π j of triplets Π j such that the corresponding direct sum Π = ⊕ j∈J Π j forms a boundary triplet for A * = ⊕ j∈J A * j were suggested in [14,55,69].