Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

We consider a compact metric graph of size $\varepsilon$, and attach to it several edges (leads) of length of order one (or of infinite length). As $\varepsilon$ goes to zero, the graph $\mathcal{G}^\varepsilon$ obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On $\mathcal{G}^\varepsilon$ we define an Hamiltonian $H^\varepsilon$, properly scaled with the parameter $\varepsilon$. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates $H^\varepsilon$ (in a suitable norm resolvent sense) as $\varepsilon\to0$. The effective Hamiltonian depends on the spectral properties of an auxiliary $\varepsilon$-independent Hamiltonian defined on the compact graph obtained by setting $\varepsilon = 1$. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit $\varepsilon\to0$, the leads are decoupled.


Introduction
One nice feature of quantum graphs (metric graphs equipped with differential operators) is that they are simple objects. In many cases, for example in the framework of the analysis of self-adjoint realizations of the Laplacian, it is possible to write down explicit formulae for the relevant quantities, such as the resolvent or the scattering matrix (see, e.g., [23] and [24]).
If the graph is too intricate though, it can be difficult, if not impossible, to perform exact computations. In such a situation, one may be interested in a simpler, effective model which captures only the most essential features of a complex quantum graph.
If several edges of the graph are much shorter then others, an effective model should rely on a simpler graph obtained by shrinking the short edges into vertices. These new vertices should keep track of at least some of the spectral or scattering properties of the shrinking edges, and perform as a black box approximation for a small, possibly intricate, network.
Our goal is to understand under what circumstances this type of effective models can be implemented. In this report we give some preliminary results showing that under certain assumptions such approximation is possible.
To fix the ideas, consider a compact metric graph G in,ε of size (total length) ε, and attach to it several edges of length of order one (or of infinite length), the leads. Clearly, when ε goes to zero, the graph obtained in this way (let us denote it by G ε ) looks like the star-graph formed by the leads joined in a central vertex. Let us denote by G out such star-graph and by v 0 the central vertex. Given a certain Hamiltonian (self-adjoint Schrödinger operator) H ε on G ε , we want to show that there exists an Hamiltonian H out on G out such that, for small ε, H out approximates (in a sense to be specified) H ε . Of course, one main issue is to understand what boundary conditions in the vertex v 0 characterize the domain of H out .
It turns out that, under several technical assumptions, the boundary conditions in v 0 are fully determined by the spectral properties of an auxiliary, ε-independent Hamiltonian defined on the graph G in = G in,ε=1 .
Below we briefly discuss these technical assumptions, and refer to Section 2 for the details.
(i) The Hamiltonian H ε on G ε is a self-adjoint realization of the operator −∆ + B ε on G ε , where B ε is a potential term. (ii) To set up the graph G ε we select N distinct vertices in G in,ε (we call them connecting vertices) and attach to each of them one lead, which is either a finite or an infinite length edge. The domain of H ε is characterized by Kirchhoff (also called standard or free) boundary conditions The author is grateful to Gregory Berkolaiko and Andrea Posilicano for enlightening discussions. The author also thanks the anonymous referees for many useful comments that helped to improve the quality of the paper. [5] there are no restrictions on the topology of the graph, i.e., G out is not necessarily a star-graph; outer edges can form loops, be connected among them or to arbitrarily intricate finite length graphs. In [5], moreover, the scale invariance assumption is missing. With respect to our work, however, the potential terms in [5] do not play an essential rôle in the limiting problem (because they are uniformly bounded in the scaling parameter).
As it was done in [5], to analyze the convergence of H ε to H out , since they are operators on different Hilbert spaces, one could use the notion of δ ε -quasi unitary equivalence (or generalized norm resolvent convergence) introduced by P. Exner and O. Post in the series of works [16,17,18,31,32]. In Th.s 2.12 and 2.13 we state our main results in terms of the expansion of the resolvent in the decomposition L 2 (G ε ) = L 2 (G out ) ⊕ L 2 (G in,ε ); and comment on the δ ε -quasi unitary equivalence of the operators H ε andH out (or H out ) in Rem. 2.15.
Our analysis, with the scaling on the potential B in,ε (x) = ε −2 B in (x/ε), is also related to the problem of approximating point-interactions on the real line through scaled potentials in the presence of a zero energy resonance, see, e.g., [20]. The same type of scaling arises naturally also in the study of the convergence of Schrödinger operators in thin waveguides to operators on graphs, see, e.g., [1,9,10,11].
Problems on graphs with a small compact core have been studied in several papers in the case in which G ε is itself a star-graph, see, e.g., [14,15,25,26,27]. In particular, in the latter series of works, the authors point out the rôle of the zero energy eigenvalue.
Also related to our work is the problem of the approximation of vertex conditions through "physical Hamiltonians". In [12] (see also references therein), it is shown that all the possible self-adjoint boundary conditions at the central vertex of a star-graph, can be obtained as the limit of Hamiltonians with δ-interactions and magnetic field terms on a graph with a shrinking inner part.
Instead of looking at the convergence of the resolvent, a different approach consists in the analysis of the time dependent problem. This is done, e.g., in [3], for a tadpole-graph as the circle shrinks to a point.
The paper is structured as follows. In Section 2 we introduce some notation, our assumptions and present the main results, see Th.s 2.12 and 2.13. In Section 3 we discuss the Kreȋn formulae for the resolvents of H ε and H out (the limiting Hamiltonian in the Non-Generic Case). These formulae are the main tools in our analysis. In Section 4 we discuss the scale invariance properties of the auxiliary Hamiltonian, and other relevant operators. In Section 5 we prove Th.s 2.12 and 2.13. In doing so we present the results with a finer estimate of the remainder, see Th.s 5.4 and 5.9. We conclude the paper with two appendices: in Appendix A we briefly discuss the proofs of the Kreȋn resolvent formulae from Section 3; in Appendix B we prove some useful bounds on the eigenvalues and eigenfunctions ofH in .
Index of notation. For the convenience of the reader we recall here the notation for the Hamiltonians used in our analysis. For the definitions we refer to Section 2 below.
•H out effective Hamiltonian in the Generic Case.
• H out effective Hamiltonian in the Non-Generic Case.

Preliminaries and main result
For a general introduction to metric graphs we refer to the monograph [4]. Here, for the convenience of the reader, we introduce some notation and recall few basic notions that will be used throughout the paper.
2.1. Basic notions and notation. To fix the ideas we start by selecting a collection of points, the vertices of the graph, and a connection rule among them. The bonds joining the vertices are associated to oriented segments and are the finite-length edges of the graph. Other edges can be of infinite length, these edges are connected only to one vertex and are associated to half-lines. In this way we obtained a metric graph, see, e.g., Fig. 1. Given a metric graph G we denote by E the set of its edges and by V the set of its vertices. We shall also use the notation |E| and |V| to denote the cardinality of E and V respectively. We shall always assume that both |E| and |V| are finite.
For any e ∈ E, we identify the corresponding edge with the segment [0, ℓ e ] if e has finite length ℓ e > 0, or with [0, +∞) if e has infinite length. Given a function ψ : G → C, for e ∈ E, ψ e denotes its restriction to the edge e. With this notation in mind one can define the Hilbert space with scalar product and norm given by and ψ H := (ψ, ψ) In a similar way one can define the Sobolev space H 2 := e∈E H 2 (e), equipped with the norm Note that functions in H 2 are continuous in the edges of the graph but do not need to be continuous in the vertices. For any vertex v ∈ V we denote by d(v) the degree of the vertex, this is the number of edges having one endpoint identified by v, counting twice the edges that have both endpoints coinciding with v (loops). Let E v ⊆ E be the set of edges which are incident to the vertex v. For any vertex v we order the edges in E v in an arbitrary way, counting twice the loops. In this way, for an arbitrary function ψ ∈ H 2 , one can define the vector Ψ(v) ∈ C d(v) associated to the evaluation of ψ in v, i.e., the components of Ψ(v) are given by ψ e (0) or ψ e (ℓ e ), e ∈ E v , depending whether v is the initial or terminal vertex of the edge e, or by both values if e is a loop. In a similar way one can define the vector Ψ ′ (v) ∈ C d(v) with components ψ ′ e (0) and −ψ ′ e (ℓ e ), e ∈ E v . Note that in the definition of Ψ ′ (v), ψ ′ e denotes the derivative of ψ e (x) with respect to x, and the derivative in v is always taken in the outgoing direction with respect to the vertex. We are interested in defining self-adjoint operators in H which coincide with the Laplacian, possibly plus a potential term.
We denote by B the potential term in the operator, so that B : G → R is a real-valued function on the graph; and denote by B e its restriction to the edge e. Additionally we assume that B is bounded and compactly supported on G.

2.2.
Graphs with a small compact core. We consider a graph G ε obtained by attaching several edges to a small compact core (a compact metric graph of size ε). We denote the compact core of the graph by G in,ε . The graph G in,ε is obtained by shrinking a compact graph G in by means of a parameter 0 < ε < 1, more precisely, we set G in,ε = εG in . (2.4) We denote by E in the set of edges of the graph G in and by E in,ε the set of edges of the graph G in,ε .
In the graph G in (or, equivalently, in G in,ε ) we select N distinct vertices that we label with v 1 , ..., v N , and refer to them as connecting vertices. We shall denote by C the set of connecting vertices. We denote by V in the set of all the remaining vertices, and call the elements of V in inner vertices (note that the set V in may be empty).
To construct the graph G ε , we attach to each connecting vertex one additional edge which can be an half-line or an edge of finite length (not dependent on ε). We shall call these additional edges outer edges and denote by E out the corresponding set of edges; obviously |E out | = N . When needed, we shall denote these edges by e 1 , ..., e N , so that the edge e j is connected to the vertex v j , j = 1, ..., N . Moreover we shall use the notation Note that if e ∈ E out is of finite length the endpoint which does not coincide with the connecting vertex is of degree one (all the finite length outer edges are pendants).
We shall always assume, without loss of generality, that for each edge in E out the connecting vertex is identified by x = 0. We denote by E ε and V the sets of edges and vertices of the graph G ε . We note that E ε = E out ∪ E in,ε and V = V out ∪ C ∪ V in , where V out is the set of vertices in G ε which are neither connecting nor inner vertices.
As ε → 0, the inner graph shrinks to one point, in the limit all the connecting vertices merge in one vertex which we identify with the point x j = 0, x j being the coordinate along the edge e j ∈ E out , j = 1, . . . , N . In the limit the graph G ε looks like a star-graph with N edges connected in the origin, see Fig. 2; we denote the star-graph by G out .
∞ ∞ ∞ Figure 2. The dashed lines represent the edges of G in,ε , the large dots the connecting vertices. The graph G out is obtained by merging the connecting vertices. In the example in the picture, G out has three infinite edges and one edge of finite length.
We define the Hilbert spaces: We remark that one can always think of H ε as the direct sum and decompose each function ψ ∈ H ε as ψ = (ψ out , ψ in ) with ψ out ∈ H out and ψ in ∈ H in,ε . When no misunderstanding is possible, we omit the dependence on ε, moreover we simply write ψ, instead of ψ out or ψ in . In a similar way we introduce the Sobolev spaces 3)); this is the object of our investigation.
• Recall that if v ∈ V out , then d(v) = 1. For any v ∈ V out we fix an orthogonal projection P out v : C → C, and a self-adjoint operator Θ out v in Ran(P out v ). Since vertices in V out have degree one, P out v is either 1 or 0; whenever P out v = 1 it makes sense to define Θ out v which turns out to be the operator acting as the multiplication by a real constant. In other words, the boundary conditions in v ∈ V out (of the form given in the definition of D(H P,Θ )) can be of Dirichlet type, ψ e (v) = 0; of Neumann type ψ ′ e (v) = 0; or of Robin type ψ ′ e (v) = αψ e (v) with α ∈ R. It would be possible to consider a more general setting in which the outer graph has a non trivial topology, in same spirit of the work [5], but we will not pursue this goal.
• For any v ∈ C we define the orthogonal projection (see Rem. 2.1 for the definition of d(v)): . In a similar way, we define the orthogonal projection For a discussion on the meaning and the main consequences of these assumptions we refer to Section 4. Definition 2.2 (Hamiltonian H ε ). We denote by H ε the self-adjoint operator in H ε defined by Remark 2.3. In the out/in decomposition one has Note that the action of the outer component of H ε does not depend on ε.
where the sum is taken on all the edges incident on v (counting loops twice) and the derivative is understood in the outgoing direction from the vertex.
2.4. Auxiliary Hamiltonian. We are interested in the limit of the operator H ε as ε → 0. We shall see that the limiting properties of H ε are strongly related to spectral properties of the HamiltonianH in,ε : Definition 2.5 (Auxiliary Hamiltonian, scaled down version).
Let H in = H in,ε=1 , and define the unitary scaling group

one infers the unitary relation
withH in defined on H ε and given byH in =H in,ε=1 . One consequence of Eq. (2.7) is that the spectrum ofH in,ε is related to the spectrum ofH in by the relation σ(H in,ε ) = ε −2 σ(H in ) (see Section 4 for more comments on the implications of the scale invariance assumption). For this reason, we prefer to formulate the results in terms of the spectral properties of the ε-independent HamiltonianH in instead of the spectral properties ofH in,ε .
Definition 2.6 (Auxiliary HamiltonianH in ). We call Auxiliary Hamiltonian the , the domain and action ofH in are given by The spectrum ofH in consists of isolated eigenvalues of finite multiplicity, see, e.g., [4, Th. 3.1.1]. For n ∈ N, we denote by λ n the eigenvalues ofH in (counting multiplicity) and by {ϕ n } n∈N a corresponding set of orthonormal eigenfunctions. Definition 2.7 (Generic/Non-Generic Case). In the analysis of the limit of H ε we distinguish two cases: (1) Generic (or Non-Resonant, or Decoupling) Case. λ = 0 is not an eigenvalue of the operatorH in .
In the Non-Generic Case we denote by {φ k } k=1,...,m a set of (orthonormal) eigenfunctions corresponding to the zero eigenvalue. By Eq.
In the Non-Generic Case, let C be the operator C is a bounded self-adjoint operator (it is an N × N Hermitian matrix). Denote by Ran C ⊆ C N and Ker C ⊆ C N , the range and the kernel of C respectively. One has that the subspaces Ran C and Ker C are C-invariant. Moreover, C N = Ran C ⊕ Ker C. In what follows we denote by P the orthogonal projection (Riesz projection, see, e.g., [19, Section I.2]) on Ran( C), and by P ⊥ = I N − P the orthogonal projection on Ker( C).
Remark 2.9. We note that q ∈ Ker C if and only if (ĉ k , q) C N = 0 for all k = 1, . . . , m. To see that this indeed the case, observe that if q ∈ Ker C then it must be (q, Cq) C N = 0, hence, m k=1 |(ĉ k , q) C N | 2 = 0, which in turn implies (ĉ k , q) C N = 0 for all k = 1, . . . , m. The other implication is trivial.
Since P ⊥ĉ k ∈ Ker C, we infer 0 = (ĉ k , P ⊥ĉ k ) C N = ( P ⊥ĉ k , P ⊥ĉ k ) C N = P ⊥ĉ k 2 C N for all k = 1, . . . , m; hence, P ⊥ĉ k = 0, or, equivalently,ĉ k ∈ Ran( C). 2.5. Effective Hamiltonians. We shall see that the definition of the limiting operator (effective Hamiltonian in H out ) depends on presence of a zero eigenvalue forH in (the occurrence of the Generic Case vs. the Non-Generic Case).
Recall that for ψ ∈ H out , we used ψ j to denote the component of ψ on the edge e j attached to the connecting vertex v j . Moreover, we assumed that the vertex v j is identified by x = 0. With this remark in mind, given a function ψ ∈ H out 2 we define the vectors These correspond to Ψ(v 0 ) and Ψ ′ (v 0 ), as defined in Section 2.1, where v 0 is the central vertex of the star-graph G out .
In the limit ε → 0, the connecting vertices in G in,ε coincide, and can be identified with the vertex v 0 ≡ 0.
We distinguish two possible effective Hamiltonians in H out .
Definition 2.10 (Effective Hamiltonian, Generic Case). We denote byH out the self-adjoint operator in H out defined by

Main result.
In what follows C denotes a generic positive constant independent on ε. Given two Hilbert spaces X and Y , we denote by B(X, Y ) (or simply by B(X) if X = Y ) the space of bounded linear operators from X to Y , and by · B(X,Y ) the corresponding norm. For any a ∈ R, we use the notation O B(X,Y ) (ε a ) to denote a generic operator from X to Y whose norm is bounded by Cε a for ε small enough.
Given a bounded operator A in H ε we use the notation

13)
where the expansion has to be understood in the out/in decomposition (2.11).
Theorem 2.13. Let z ∈ C\R. In the Non-Generic Case (see Def. 2.7), let C 0 be the restriction of C to Ran P . (i) If Ker C ⊂ C N , C 0 is invertible as an operator in P C N , and (2.14) where the expansion has to be understood in the out/in decomposition (2.11). (ii) If Ker C = C N , then P = 0, and expansion (2.14) holds true with H out =H out , (ĉ k , C −1 0ĉk ′ ) C N = 0 for all k, k ′ = 1, . . . , m, and the error term changed in O B(H ε ) (ε).
(iii) If the vectorsĉ k , k = 1, . . . , m, are linearly independent, then δ k,k ′ − (ĉ k , C −1 0ĉk ′ ) C N = 0 for all k, k ′ = 1, . . . , m, and where (ψ out , 0) is understood in the decomposition (2.5). Its adjoint J * maps H ε in H out , and is given by: Note that J * J = I out , where I out is the identity in H out . The operator H ε is δ ε -quasi unitarily equivalent to a self-adjoint operator H out in H out if for some z ∈ C\R. Note that in the decomposition (2.12), one has Hence: By Th. 2.12, in the Generic Case the operator H ε is ε-quasi unitarily equivalent to the operatorH out . By Th. 2.13 -(iii), in the Non-Generic Case, if the vectorsĉ k , k = 1, . . . , m, are linearly independent, the operator H ε is ε 1/2 -quasi unitarily equivalent to the operator H out . More precisely, the second condition in Eq. (2.16) always holds true, while the first one holds true only under the additional assumption that the vectorsĉ k are linearly independent.
We refer to [32] for a comprehensive discussion on the comparison between operators acting on different spaces.

Kreȋn resolvent formulae
In this section we introduce the main tools in our analysis: the Kreȋn-type resolvent formulae for the resolvents of H ε and H out . The proofs are postponed to App. A.
Given the Hilbert spaces X out , Y out , X in , and Y in , and a couple of operators A out : X out → Y out and A in : X in → Y in , we denote by A := diag(A out , A in ), the operator A : X → Y , with X := X out ⊕ X in and Y := Y out ⊕ Y in , acting as Af := (A out f out , A in f in ), for all f = (f out , f in ) ∈ X, f out ∈ X out and f in ∈ X in . We set D(H ε ) := D(H out ) ⊕ D(H in,ε ) andH ε := diag(H out ,H in,ε ), (3.1) withH out andH in,ε given as in Def.s 2.10 and 2.5 Given an operator A, we denote by ρ(A) its resolvent set; the resolvent of A is defined as (A − z) −1 for all z ∈ ρ(A).
For the resolvents of the relevant operators we introduce the shorthand notation (3.5) Obviously, all the operators in Eq.s. (3.2) -(3.5) are well-defined and bounded for z ∈ C\R, moreover Our aim is to write the resolvent difference R ε z −R ε z in a suitable block matrix form, associated to the off-diagonal matrix Θ in Eq. (3.13). To do so we follow the approach of Posilicano [29,30]. All the self-adjoint extensions of the symmetric operator obtained by restricting a given self-adjoint operator to the kernel of a given map τ are parametrized by a projection P and a self-adjoint operator Θ in Ran P. We choose the reference operatorH ε and the map τ so that the Hamiltonian of interest H ε is the self-adjoint extension parametrized by the identity and the self-adjoint operator given by the off-diagonal matrix Θ. The Kreȋn formula for the resolvent difference R ε z −R ε z , see Lemma 3.2, is obtained within the approach from [29,30].
We define the maps: Moreover we set, τ : H ε 2 = H out 2 ⊕ H in,ε . Recall that we have denoted by λ n and {ϕ n } n∈N the eigenvalues and a corresponding set of orthonormal eigenfunctions ofH in .
The eigenvalues ofH in,ε (counting multiplicity) and a corresponding set of orthonormal eigenfunctions are given by λ ε n = ε −2 λ n ; ϕ ε n (x) = ε −1/2 ϕ n (x/ε), (4.1) where λ n are the eigenvalues ofH in , and ϕ n the corresponding (orthonormal) eigenfunctions. By the spectral theorem and by the scaling properties (4.1),R in,ε z is given bẙ Hence, its integral kernel can be written as Since there exists a positive constant C such that sup x∈G in |ϕ n (x)| ≤ C and λ n ≥ Cn 2 for n large enough (see App. B), the series in Eq. (4.3) is uniformly convergent for x, y ∈ G in,ε . Hence, we can write the operatorsG in,ε z and G in,ε z , and the matrix M in,ε z in a similar way, see Eq.s (4.4) and (4.5) below. Note that, since functions in D(H in,ε ) are continuous in the connecting vertices, the eigenfunctions ϕ ε n can be evaluated in the connecting vertices, and, by the definition of τ in (see Eq. (3.7)), one has τ in ϕ ε n = (ϕ ε n (v 1 ), . . . , ϕ ε n (v N )) T .
So that, for any eigenfunction ϕ ε n we can define the vector c ε n as c ε n := τ in ϕ ε n . We note that c ε n = ε −1/2 c n , with c n = (ϕ n (v 1 ), . . . , ϕ n (v N )) T , and that the vectors c n are defined in the same way as the vectorsĉ k in Eq. (2.9).
To prove the first claim in Eq. (5.7), let ψ in ∈ H in,ε , then Hence, from the Cauchy-Schwarz inequality, This proves the first Claim in Eq. (5.7); the second one is trivial, being G in,ε z the adjoint ofG in,ε z . Proposition 5.3. Let z ∈ C\R. In the Generic Case, Proof. Recall Eq. (4.5) and note that for any q ∈ C N , Proof. We prove first Claim (5.10). By Eq. (4.2) we infer Note that the second sum runs over λ n = 0, hence one has the bound |λ n − ε 2 z| ≥ |λ n |/2 ≥ C, for ε small enough. For this reason, the bound in Eq. (5.10) on the second term at the r.h.s. of Eq. (5.13) can be obtained with an argument similar to the one used in the proof of bound (5.6).
To prove Claim (5.11) we proceed in a similar way. We note that, see Eq. (4.4), The latter can also be written as Which, together with Eq. (5.23), allows us to infer the expansion and conclude the proof of the proposition.
We are now ready to state and prove the main theorem for the Non-Generic Case. In the statement of the theorem, we assume that Ker C ⊂ C N , i.e., P = 0. In this way the quantity (ĉ k , C −1 0ĉk ′ ) C N is certainly well defined. We discuss the case Ker C = C N (i.e., P = 0) separately in the proof of point (ii) of Th. 2.13 (after the proof of Th. 5.9).
Theorem 5.9. Let z ∈ C\R. In the Non-Generic Case assume that Ker C ⊂ C N , then Taking into account Eq. (5.25) and the expansion (5.12), we infer that, for all ψ ∈ H in,ε the leading term in R in,in,ε z ψ is given by the remainder being of order ε. From the latter formula and from the expansion (5.10) we infer Th. 2.13 -(i) follows immediately from Th. 5.9.
Proof of Th. 2.13 -(ii). If Ker C = C N thenĉ k = 0, for all k = 1, . . . , m, see Rem. 2.9. Hence, expansions (5.11), (5.12), and (5.14) read respectively Reasoning along the lines of the analysis of the Generic Case, see the proof of Th. 5.4, and taking into account the expansion (5.10), one readily infers which implies the statement in Th. 2.13 -(ii).

Proof of Th. 2.13 -(iii).
To prove the second part of Th. 2.13, recall thatĉ k ′ ∈ P C N and C −1 If the vectors {ĉ k } m k=1 are linearly independent this linear combination is zero if and only if δ k,k ′ − (ĉ k , C −1 0ĉk ′ ) = 0 for all k. Hence, expansion (2.15) follows from Eq. (2.14). Remark 5.10. Denote by Λ the operator in H in,ε defined by Λ is selfadjoint and Λ 2 = Λ. The first claim is obvious (recall that C 0 is selfadjoint). To prove the second claim, note that, since (φ ε l ′ , ϕ ε k ) H in,ε = δ l ′ ,k , Hence, Λ is an orthogonal projection in H in,ε . and R ε z are the resolvents of a self-adjoint extension of the symmetric operatorsH out ↾ Kerτ out andH ε ↾ Kerτ respectively.
We are left to prove that such self-adjoint extensions coincide with H out and H ε respectively. Let us focus attention on R ε z (similar considerations hold true for R out z ). Since the self-adjoint operator associated to R ε z is an extension ofH ε ↾ Kerτ , to prove that R ε z is the resolvent of H ε , we just need to check that in the connecting vertices functions in Ran R ε z satisfy the boundary conditions required by D(H ε ). The remaining boundary conditions are clearly satisfied because the mapτ evaluates functions only in the connecting vertices.
Define the maps: . We recall the following formula which is obtained by integrating by parts For all φ ∈ D(H ε ) and ψ as above, the identity (A.4) gives In what follows we use the decomposition C 2N = C N ⊕ C N , so that q = (q out , q in ) and τ φ = (τ out φ out , τ in φ in ).
Next let φ = (0, φ in ). Then Identity (A.5) gives In a similar way one can prove q in j = (σ in ψ in ) j , j = 2, . . . , N , hence, σ in ψ in = q in . We also note that the function ψ is continuous in the connecting vertices (whenever the vertex degree is larger or equal than two). To see that this is indeed the case, consider in Eq. (A.7) a function φ in such that φ in (v j ) = 0, j = 1, . . . , N , Φ in ′ (v 1 ) = (1, −1, 0, . . . , 0) T := e, Φ in ′ (v j ) = 0, j = 2, . . . , N . Since K in⊥ v1 e = e, condition (A.7) gives (e, Ψ in (v 1 )) = 0. Repeating the process, moving −1 in the vector e on all the positions (from the second one on) one obtains the continuity of ψ in the vertex v 1 . The same holds true for every connecting vertex.
We have proved that for any χ ∈ H ε , setting q = M ε z − Θ −1G ε z χ ∈ C 2N , one has: σ out G out z q out = q out ; σ in G in,ε z q in = q in ; σG ε z q = q. (A.8)

Appendix B. Estimates on eigenvalues and eigenfunctions ofH in
In this appendix we prove the following proposition on the asymptotic behavior of eigenvalues and eigenfunctions ofH in .
Proposition B.1. Recall that we denoted by {λ n } n∈N the eigenvalues of the HamiltonianH in , and by {ϕ n } n∈N a corresponding set of orthonormal eigenfunctions. There exists n 0 such that for any n ≥ n 0 : λ n > n 2 C (B.1) and sup x∈G in for some positive constant C which does not depend on n.
Proof. Claim (B.1) is just the Weyl law. For B in = 0 a proof can be found in [6,Prop. 4.2] (see also [28]). For B in = 0 bounded, claim (B.1) can be deduced by a perturbative argument.
To prove the bound (B.2) we follow the lines in the proof of Theorem A.1 in [13]. For b ∈ L ∞ (0, ℓ) and real valued, and λ > 0 let f be the solution of the equation where C is a positive constant which does not depend on λ, f 0 and f ′ 0 . We have then proved that Any component of the eigenfunction ϕ n satisfies in the corresponding edge an equation of the form (B.3) with some initial data in x = 0. Then the discussion on the function f (x) above applies to all the components of the vector ϕ n . By the normalization condition ϕ n H in = 1 it follows that it must be f L 2 ((0,l)) = C, with C ≤ 1 (here f denotes a generic component of ϕ n , i.e., the restriction of ϕ n to a generic edge of G in ). Hence, from the identity one infers The latter estimate implies that there existsλ such that, for all λ >λ, the inequalities |f 0 | ≤ C 1 and |f ′ 0 |/ √ λ ≤ C 1 hold true for some positive constant C 1 which does depend on λ. The bounds |f 0 | ≤ C 1 and |f ′ 0 |/ √ λ ≤ C 1 , together with estimate (B.6) and the fact that λ n → +∞ for n → ∞, imply (B.2).