The Krein-von Neumann extension for Schr\"odinger operators on metric graphs

The Krein-von Neumann extension is studied for Schr\"odinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.


Introduction
It is an almost hundred-year-old story that many of the differential operators appearing in mathematical physics and their boundary conditions can be described conveniently in the framework of extension theory of symmetric operators. A complete description of all self-adjoint extensions of a symmetric operator was first given by von Neumann [40]. On the other hand, it turned out that a theory of self-adjoint extensions of symmetric operators that are semibounded from below can be done conveniently by means of semibounded sesquilinear forms; this originates from the work of Friedrichs [21]. However, it is due to Krein [32] (see also the works of Vishik [49] and Birman [17]) that among all non-negative extensions of a positive definite symmetric operator S, there are two extremal ones, the Friedrichs extension S F and the (by now so-called) Krein-von Neumann extension S K , in the sense that each non-negative self-adjoint extension A of S satisfies These inequalities may be understood in the sense of quadratic forms or via the involved operators' resolvents. It is beyond the scope of this article to provide a complete historical review of the developments related to the Krein-von Neumann extension; for further reading we refer the reader to [2] and the survey articles [1,8]. Among the abstract advancements on extremal extensions of positive definite symmetric operators (and, more generally, symmetric linear relations), we mention [3,5,6,18,25,36,42,46,47,48].
In the study of e.g. elliptic second order differential operators on Euclidean domains, the Friedrichs extension is a very natural object; for instance, for the minimal symmetric Laplacian on a bounded domain in R n corresponding to both Dirichlet and Neumann boundary conditions, the Friedrichs extension is the self-adjoint Laplacian subject to Dirichlet boundary conditions. On the other hand, in the same setting, the Krein-von Neumann extension corresponds to certain non-local boundary conditions which can be described in terms of the associated Dirichletto-Neumann map; for properties of the Krein-von Neumann extension of elliptic differential operators and recent related developments, we refer the reader to [4,9,11,23,24,37]. 1 For differential operators on metric graphs, which we consider in the present paper, the situation is similar, yet different in some respects. If Γ is a finite metric graph, then we take, as a starting point, the (negative) Laplacian (i.e. the negative second derivative operator on each edge) S in L 2 (Γ), which satisfies on each vertex both Dirichlet and Kirchhoff vertex conditions; that is, the functions in the domain of S vanish and have derivatives which sum up to zero at each vertex. This symmetric operator is very natural to carry out extension theory, since its adjoint S * is the Laplacian on Γ with continuity as its (only) vertex conditions. Therefore any self-adjoint extension of S in L 2 (Γ) (which, at the same time is a restriction of S * ) satisfies continuity conditions and thus reflects, at least to some extent, the connectivity of the graph. Nevertheless, the Friedrichs extension of S in this setting is the Laplacian on functions which are zero at every single vertex, an operator which, despite continuity, is determined by the graph's edges considered as separate intervals, instead of the actual graph structure. Other self-adjoint extensions of S which are more suitable for the spectral analysis of network structures are the operator with continuity-Kirchhoff conditions (the so-called standard or natural Laplacian) or with δ-type vertex conditions; the latter we will not discuss here further.
The Krein-von Neumann extension for a Laplacian on a metric graph has not been considered much in the literature so far; an attempt for a symmetric operator with vertex conditions different from the ones considered here was done in [38]. The Krein-von Neumann extension of our operator S is, like for the minimal Laplacian on a Euclidean domain, an operator with non-local vertex conditions. Nevertheless, its domain is intimately connected to the structure of the underlying graph. In fact, we prove that the matrix that couples the values and the sums of derivatives at the vertices for functions in the domain of S K is exactly the weighted discrete Laplacian on the underlying discrete graph, where the weights are the inverse edge lengths.
Our main focus in the present paper is on spectral properties of the operator S K , not only in the case of the Laplacian, but also for Schrödinger operators with nonnegative potentials q e on the edges. Namely, we consider the operator S acting as − d 2 dx 2 + q e on each edge e of Γ, with Dirichlet and Kirchhoff vertex conditions as described above in the case of the Laplacian. Its Krein-von Neumann extension, the so-called perturbed Krein Laplacian, denoted by −∆ K,Γ,q , is the main object of consideration in this article. We first describe the domain of −∆ K,Γ,q in terms of vertex conditions and establish Krein-type formulae for the resolvent differences with both the Friedrichs extension (the Schrödinger operator with Dirichlet vertex conditions) and the the Schrödinger operator −∆ st,Γ,q with standard vertex conditions. As a consequence, we obtain the formula in which V denotes the number of vertices of Γ and λ takes appropriate complex values. This formula distinguishes the potential-free case clearly from the case influenced by a potential. It also sheds light on another interesting phenomenon: the Krein Laplacian may, in some rare occasions, coincide with the standard Laplacian, and this is the case if and only if Γ has only one vertex (with possibly many loops attached to it) and thus is a so-called flower graph. Moreover, we use the Krein-type resolvent formulae to obtain some results on spectral asymptotics of the perturbed Krein Laplacian. A further property of the perturbed Krein Laplacian on a metric graph Γ, which we establish, is the possibility to describe its positive eigenvalues variationally. In fact, the spectrum of −∆ K,Γ,q is purely discrete, and the lowest eigenvalue is always zero, with multiplicity equal to V , the number of vertices, as we show. All its positive eigenvalues λ + j (−∆ K,Γ,q ), ordered nondecreasingly and counted with multiplicities, can be characterised by the variational principle here, H 2 0 (Γ) is the second-order Sobolev space on each edge, equipped with Dirichlet and Kirchhoff conditions on all edges. This formula is the exact counterpart of a variational description of the positive eigenvalues of the perturbed Krein Laplacian on a domain in R n , which was established in [9,Proposition 7.5]. Before we derive (1.1), we first establish an abstract version of this principle; see Theorem 2.4. Its proof is along the lines of the result for the Laplacian in [9]; however, we found it useful and of independent interest to have it at hand also abstractly for the Krein-von Neumann extension of any symmetric, positive definite operator S for which dom S equipped with the graph norm of S satisfies a compactness condition. As a consequence of the formulation for graphs (1.1), we easily obtain inequalities between the (positive) eigenvalues of the perturbed Krein Laplacian and other self-adjoint extensions of S.
An important field of application of the eigenvalue characterisation (1.1) are so-called surgery principles. Such principles study the influence of geometric perturbations of a metric graph on the specta of associated Laplacians or more general differential operators. The reader may think of sugery operations such as joining two vertices into one or cutting through a vertex, or adding or removing edges (or even entire subgraphs). Such principles were studied in depth for the Laplacian or Schrödinger operators subject to standard (and some other local) vertex conditions; see [15,26,29,34,44]. As we point out, the eigenvalues of the perturbed Krein Laplacian behave in some respects in the same way as the eigenvalues of −∆ st,Γ,q ; for instance, when gluing vertices all eigenvalues increase (or stay the same), and adding pendant edges or graphs (a process which increases the "volume" of Γ) may only decrease the eigenvalues. On the other hand, in some respects the behaviour is different from what we are used to for standard vertex conditions. Let us only mention three examples: firstly, for the positive eigenvalues, gluing vertices has actually a non-increasing effect (but at the same time also the multiplicity of the eigenvalue 0 decreases), whilst for standard vertex conditions, the positive eigenvalues behave non-decreasingly and the dimension of the kernel remains the same. Secondly, removing a vertex of degree two (replacing the two incident edges by one) may change eigenvalues in a monotonous way, whilst it does not have any influence on the spectrum of an operator with standard vertex conditions. Thirdly, inserting an edge between two existing vertices makes all eigenvalues decrease (or stay the same); for standard vertex conditions, this is not necessarily the case; see e.g. [33].
A typical application of surgery principles for graph eigenvalues consists of deriving spectral inequalities in terms of geometric and topological parameters of the graph such as its total length, diameter, number of edges or vertices, or its first Betti number (or Euler characteristics, equivalently). For a few recent advances on spectral inequalities for quantum graphs, we refer to [10,14,28,31,39,41]. To demonstrate how surgery principles for the perturbed Krein Laplacian on a metric graph may be applied, we establish lower bounds for the positive eigenvalues, in terms of eigenvalues of a loop graph or edge lengths. For instance, for the first positive eigenvalue of the Krein Laplacian without potential the lower bound is explicit, where ℓ(Γ) denotes the total length of Γ, and we specify the class of graphs for which this estimate is optimal.
Considering the Krein-von Neumann (and other) extensions of a Schrödinger operator with Dirichlet and Kirchhoff vertex conditions at all vertices is natural, as we pointed out above. However, it may also be useful to study extensions of a symmetric Schrödinger operator with different vertex conditions. We mention, as an example, the Laplacian with both Dirichlet and Neumann (Kirchhoff) vertex conditions at the "loose ends", i.e. the vertices of degree one, but standard vertex conditions at all interior vertices. In this case, the vertex conditions of the Kreinvon Neumann extension will still be standard at all interior vertices, but they will couple the vertices of degree one in a nonlocal way. We conclude our paper with a short section where we discuss such situations.
Let us briefly describe how this paper is organised. In Section 2, we review some background on the abstract Krein-von Neumann extension. Moreover, we provide a proof of the abstract counterpart of the variational principle (1.1) and derive a few easy consequences. Additionally, we study some basic properties of boundary triples, which we use as a tool. The aim of Section 3 is to introduce the perturbed Krein Laplacian on a metric graph and to study its properties, such as a description of its domain, Krein-type resolvent formulae and some consequences of the min-max principle. Section 4 is devoted to a collection of surgery principles, whilst in Section 5, we apply some of them in order to obtain some isoperimetric inequalities. Finally, Section 6 deals with the more general setting where self-adjoint vertex conditions are fixed at some vertices, and extension theory is applied with respect to the remaining vertices.
2. The abstract Krein-von Neumann extension and its eigenvalues 2.1. Preliminaries. Throughout this section we assume that H is a separable complex Hilbert space with inner product (·, ·) and corresponding norm · . For any closed linear operator A in H, we denote by σ(A) and ρ(A) its spectrum and resolvent set respectively. If A is self-adjoint and has a purely discrete spectrum, then we write for its eigenvalues, counted according to their multiplicities. If G is a further Hilbert space, we denote by B(G, H) the space of all bounded, everywhere-defined linear operators from G to H and abbreviate B(G) := B(G, G).
We make the following assumption.
Under Hypothesis 2.1, the defect numbers (n − , n + ) of S satisfy n − = n + = dim ker S * , where S * denotes the adjoint of S. Moreover, dom S ∩ ker S * = {0} and the Krein-von Neumann extension of S can be defined as follows.
Definition 2.2. The Krein-von Neumann extension of S is the operator S K in H given by It is well-known that S K is self-adjoint and is the smallest non-negative selfadjoint extension of S in the sense of quadratic forms. Its counterpart, the Friedrichs extension of S, is the largest non-negative extension of S and we denote it by S F . It can be defined via completion of the quadratic form induced by S; we do not go into the details but refer the reader to, e.g. the discussion in [27,Chapter VI]. For any self-adjoint, non-negative extension A of S, the relation holds for each λ < 0. The spectrum of the Friedrichs extension has a strictly positive lower bound; in fact, min σ(S F ) coincides with the supremum over all µ such that (2.1) holds. Conversely, the Krein-von Neumann extension S K has the point 0 as the bottom of its spectrum, and the corresponding eigenspace is given by which follows from the definition of S K and the fact that 0 is not an eigenvalue of S. In particular, dim ker S K = n − = n + , the defect number of S. We refer the reader to, e.g. the survey [8] for a more detailed discussion of the Krein-von Neumann extension.

2.2.
A variational characterisation of the positive eigenvalues of the Kreinvon Neumann extension. The main goal of this subsection is to provide an abstract variational description of the eigenvalues different from 0 of the Krein-von Neumann extension. The credits for the arguments that lead to the min-max principle in Theorem 2.4 below go to the articles [7,8,9], where the abstract Krein-von Neumann extension and the perturbed Krein Laplacian on domains in R n were studied. There, the min-max principle is stated in the context of the application, so for the convenience of the reader we state and prove this variational principle here abstractly.
Associated with the operator S is the space Due to (2.1), H S is a normed space, and as S is closed, it follows that H S is a Banach space. The norm · S corresponds to the inner product (f, g) S = (Sf, Sg); hence H S is a Hilbert space. Moreover, there exists a constant µ > 0 such that (Indeed, if not then for each n ∈ N there exists f n ∈ H S , w.l.o.g. f n = 1, such that Sf n < 1 n and hence µ ≤ (Sf n , f n ) ≤ Sf n < 1 n by (2.1), a contradiction to µ > 0.) We further denote by H * S the dual space of H S and write (·, ·) H * S ,HS for the sesquilinear duality between H * S and H S , i.e. the continuous extension of (h, f ) H * S ,HS : ) It will sometimes be useful to consider S as an operator from H S to H rather than as an operator in H. Therefore we define Then S is bounded and its adjoint S * is the unique bounded operator from H to H * S that satisfies Sf, g = f, S * g HS,H * S , f ∈ H S , g ∈ H.
Note that on the left-hand side we might as well replace S by S. For later use, we remark also that S * g ∈ H implies g ∈ dom S * and S * g = S * g. In particular, The following lemma is a variant of [7,Lemma 3.1]. For the convenience of the reader, we provide a complete proof. Let us show next that B is symmetric and thus self-adjoint. Indeed, for f ∈ H S , we get by (2.1) and, in particular, (Bf, f ) S ∈ R. Hence B is self-adjoint and non-negative, and (2.6) also implies that ker B = {0}. Now let λ > 0 be such that S K g = λg holds for some g ∈ dom S K , g = 0. Define also f := S −1 F S K g, where S F is the Friedrichs extension of S. As 0 / ∈ σ(S F ) by (2.1), f is well-defined and belongs to dom S F . Moreover, as g ∈ dom S K , by (2.2) we can write g = g S + g * with g S ∈ dom S and g * ∈ ker S * and get Furthermore, f = 0 as otherwise g ∈ ker S K , contradicting S K g = λg = 0, and Sf = S F f = S K g = λg together with (2.7) yields S * Sf = λ S * g = λS * (g S + g * ) = λSg S = λSf.
We point out that Lemma 2.3 describes, in an abstract setting, the coincidence between the positive eigenvalues of the Krein-von Neumann extension and the eigenvalues of an abstract buckling problem; the latter reads S * Sf = λSf and is discussed in detail in [7, Section 3].
Next we provide an abstract version of the min-max principle established for Krein Laplacians on domains in [9,Proposition 7.5]. The Rayleigh quotient is well-defined due to (2.1).
for all j ∈ N.
Proof. As the embedding ι : H S → H is compact, it follows that the Friedrichs extension S F of S has a compact resolvent, from which it can be deduced that σ(S K ) \ {0} is purely discrete; see, e.g., [8,Theorem 2.10].
For the rest of this proof, we make the abbreviation λ j := λ + j (S K ). Let B : H S → H S be the bounded, self-adjoint, nonnegative operator in Lemma 2.3 whose eigenvalues coincide with {λ −1 j : j ∈ N}. As ι is compact, the same holds for the embedding ι * : H → H * S , and B can be rewritten as which is also then compact. In particular, we can choose an orthonormal basis (Here we are assuming dim H S = ∞; the finite-dimensional case is exactly the same with a finite orthonormal basis.) Then for each j ∈ N, holds. Let us define F 0 := {0} and and denote by F ⊥ j−1 the orthogonal complement of F j−1 with respect to the inner product (·, ·) S in H S for all j ∈ N. Now fix j ∈ N. Then any f ∈ F ⊥ j−1 can be written as f = ∞ k=j c k f k for appropriate c k ∈ C, where the sum converges in H S (and hence also in H due to (2.3)). Then the continuity of S with respect to the norm in H S implies By a similar calculation, one verifies Together with (2.9), this implies the assertion of the theorem.
As a direct consequence, one gets the following comparison principle for the positive eigenvalues of S K and the eigenvalues of any self-adjoint extension of S. The inequality between eigenvalues of S F and S K is mentioned for completeness, but it has been known for a long time, see, e.g. [1, Theorem 5.1]. However, it follows conveniently from the above min-max principle.
Theorem 2.5. Assume that Hypothesis 2.1 is satisfied and that the embedding ι : H S → H is compact, and let A be any self-adjoint extension of S with a purely discrete spectrum. Moreover, let d := dim ker A. Then holds for all j ∈ N. In particular, Then for any f ∈ F and g ∈ ker A we have and hence Due to (2.1), ker A∩dom S = {0} and, thus dim(F +ker A) = j+d. Therefore (2.12) together with the usual min-max principle for A implies the assertion (2.10). Note that by the compactness of the embedding ι, the spectrum of S F is purely discrete, and thus (2.10) implies (2.11). Finally, assume that λ j (S F ) is not an eigenvalue of S, and let g = 0 in the estimate (2.12). Assuming λ j (S F ) = λ + j (S K ) for a contradiction, we get equality in (2.12) Remark 2.6. We wish to point out that compactness of the embedding of H S into H does not imply that all self-adjoint extensions of S have a purely discrete spectrum. An example is the Krein-von Neumann extension of the Laplacian with both Dirichlet and Neumann boundary conditions on a bounded, sufficiently smooth domain in R m , m ≥ 2, where ker S K = ker S * consists of all harmonic functions, and thus is infinite-dimensional, see, e.g. [9] for more details.
Remark 2.7. If the Krein-von Neumann extension of S has purely discrete spectrum (in particular d = dim ker S K is finite) we may choose A = S K in Theorem 2.5. As λ j+d (S K ) = λ + j (S K ), this shows that the inequality (2.10) is not necessarily strict in general, not even if S does not have any eigenvalues.
Given two symmetric operators S, S in H such that S ⊂ S, we get the following interlacing properties of the positive eigenvalues of their respective Krein-von Neumann extensions. We will apply it several times in subsequent sections. Theorem 2.8. Let S, S be closed, densely defined, symmetric operators in H with S ⊂ S such that (2.1) holds for S replaced by S. Moreover, assume that the embedding ι : H S → H is compact, and denote by S K and S K the Krein-von Neumann extensions of S and S respectively. Then σ(S K ) \ {0} and σ( S K ) \ {0} are purely discrete. If we assume, in addition, that dom S is a subspace of dom S of co-dimension k, then the positive eigenvalues of S K and S K satisfy the interlacing inequalities Proof. Firstly, the assumption S ⊂ S implies H S ⊂ H S algebraically, together with Hence (2.1) follows also for S, and compactness of the embedding ι implies compactness of the embedding ι : H S → H. With the help of the latter, the discreteness statement on the spectra of S K and S K follows from Theorem 2.4. Secondly, the first and third inclusion in (2.13) follow directly from the inclusion S ⊂ S, and the min-max principle in Theorem 3.11. It remains to prove the middle inequality in (2.13).
Let j ∈ N and let F ⊂ dom S be any (j + k)-dimensional subspace of dom S such that As dom S is a subspace of dom S of co-dimension k, the subspace F := F ∩ dom S of dom S satisfies dom F ≥ j, and we have which completes the proof.
We conclude this subsection with a comment on additive perturbations of the Krein-von Neumann extension. Remark 2.9. Assume that Q = Q * is a bounded, nonnegative, everywhere defined operator in H. If S is closed, symmetric, densely defined, and satisfies (2.1) then all these properties are also true for S + Q, and thus S + Q has a Krein-von Neumann extension which we denote by (S + Q) K . It is remarkable that this operator does not coincide with S K + Q, the additively perturbed Krein-von Neumann extension of S. This is in contrast to the Friedrichs extension, for which (S + Q) F = S F + Q holds. For instance, if Q = I is the identity operator then (S + I) K has a nontrivial kernel (coinciding with ker(S * + I)), whilst S K + I is bounded from below by one. Nevertheless, S K + Q is a self-adjoint, nonnegative extension of S + Q and we know thus that holds for all j ∈ N. On the other hand, by our Theorem 2.5 one has 2.3. The Krein-von Neumann extension in the framework of boundary triples. In this subsection, we review properties of the Krein-von Neumann extension in the framework of boundary triples. Our main focus is on a Krein-type formula that expresses the resolvent difference between the Krein-von Neumann extension and another self-adjoint extension of S (as, e.g. the Friedrichs extension) in terms of abstract boundary operators. We assume Hypothesis 2.1 throughout. First we recall the definition of a boundary triple.
Definition 2.10. Assume Hypothesis 2.1. A triple {G, Γ 0 , Γ 1 } consisting of a Hilbert space (G, (·, ·) G ) and two linear mappings Γ 1 , Γ 2 : dom S * → G is called boundary triple for S * if the following conditions are satisfied: We remark that boundary triples exist for any symmetric, densely defined operator S with equal defect numbers, even without the requirement (2.1). For a detailed review on boundary triples and literature references we refer the reader to, e.g. the recent monograph [12] or [45,Chapter 14].
For any given boundary triple, we have S * ↾ (ker Γ 0 ∩ ker Γ 1 ) = S, and two self-adjoint extensions of S are especially distinguished, namely (2.14) A boundary triple comes with two operator-valued functions defined on the resolvent set ρ(A) of A. defined as for f ∈ ker(S * − λ) are called γ-field and Weyl function respectively, associated with the boundary triple {G, Γ 0 , Γ 1 }.
The well-definedness of γ(λ) and M (λ) is due to the direct sum decomposition The operator γ(λ) can be viewed as an abstract Poisson operator, and M (λ) may be interpreted as an abstract Dirichlet-to-Neumann map. It is well-known that λ → M (λ) is an operator-valued Herglotz-Nevanlinna-Pick function. In particular, M (λ) is self-adjoint for λ ∈ ρ(A) ∩ R (if such points exist, which is always the case if (2.1) is assumed). Boundary triples can be used to characterise e.g. self-adjoint extensions of S in terms of abstract boundary conditions of the form Γ 1 f = ΘΓ 0 f with a selfadjoint parameter Θ acting in G. In order to actually describe all self-adjoint extensions of S, one needs to allow not only self-adjoint operators Θ but so-called self-adjoint linear relations (or multi-valued linear operators), and we do not go into these details here. For us it is sufficient to know the following; see e.g. [ the corresponding γ-field and Weyl function respectively, and A is defined in (2.14), then the following assertions hold.

holds.
Characterisations analogous to item (i) in the previous theorem hold for other types of spectra too, such as the continuous or residual spectrum, but this is not of relevance for us in this work.
If the boundary triple is chosen such that 0 ∈ ρ(A), then the Krein-von Neumann extension of S can be characterised in the following way; this is well-known, but for the convenience of the reader we repeat the short proof.
Proof. Since M (0) is self-adjoint, the restriction of S * to all f which satisfy Γ 1 f = M (0)Γ 0 f is a self-adjoint extension of S by Proposition 2.12. Moreover, by definition, each f ∈ dom S K can be written uniquely as f = f S + f * with f S ∈ dom S and f * ∈ ker S * , and therefore where we have used dom S = ker Γ 0 ∩ ker Γ 1 . This completes the proof. Now that we have this characterisation of the domain of S K at hand, we may use the above Krein-type resolvent formula to express the difference to both the distinguished self-adjoint extensions A and B of S. Proposition 2.14. Assume that Hypothesis 2.1 holds. Let {G, Γ 0 , Γ 1 } be a boundary triple for S * , and let λ → γ(λ) and λ → M (λ) denote the corresponding γ-field and Weyl function respectively. Let A and B be given in (2.14), and assume that 0 ∈ ρ(A). Then the following identities hold. holds.
Proof. Assertion (i) follows directly from plugging the result of Proposition 2.13 into the resolvent formula of Proposition 2.12 (iii). On the other hand, the operator B corresponds to the operator A Θ with Θ = 0, and hence for all λ ∈ ρ(A)∩ρ(B). For those λ which additionally belong to ρ(S K ), we combine the latter formula with assertion (i) of the present proposition to get From this, the assertion (ii) follows by an easy calculation left to the reader.
The resolvent formulae in the previous proposition may be used to determine the rank of the resolvent differences as follows.

Perturbed Krein Laplacians on metric graphs
In this section and all sections which follow, we assume that Γ is a metric graph consisting of a vertex set V, an edge set E, and a length function ℓ : E → (0, ∞) which assigns a length to each edge. Every edge e ∈ E is identified with the interval [0, ℓ(e)], and this parametrisation gives rise to a natural metric on Γ. We will always assume that Γ is finite, i.e. V := |V| and E := |E| are finite numbers, and we consider only connected graphs.
We view a function f : Γ → C as a collection of functions f e : (0, ℓ(e)) → C, e ∈ E, and say, accordingly, that f belongs to L 2 (Γ) if f e ∈ L 2 (0, ℓ(e)) for each e ∈ E. In order to define Schrödinger operators on metric graphs we make use of the Sobolev spaces H k (Γ) := f ∈ L 2 (Γ) : f e ∈ H k (0, ℓ(e)) for each e ∈ E , k ∈ N. For functions in H 1 (Γ), we may talk about continuity at a vertex v, meaning that for any two edges e,ê incident with v, the limit values (or traces) of f e and fê at the endpoints of the edges corresponding to v coincide. In this sense, we make use of the function space Moreover, for f ∈ H 2 (Γ) and v ∈ V, we write where the sum is taken over all edges e incident with v, and ∂f e (v) is the derivative of f e at the endpoint corresponding to v, taken in the direction pointing towards v; if e is a loop then both endpoints have to be taken into account.
We will consider Schrödinger operators on metric graphs with potentials that are, for simplicity, bounded. However, everything may be extended easily to formbounded (i.e. L 1 ) potentials. We will always assume the following hypothesis.
It is easy to see that S is a symmetric, nonnegative, densely defined operator in the Hilbert space L 2 (Γ). Since ⊕ e∈E C ∞ 0 (0, ℓ(e)) ⊂ dom S, the Friedrichs extension of S is the operator −∆ D,Γ,q , called the perturbed Dirichlet Laplacian, given by if q = 0 identically, we just write −∆ D,Γ and call it the Dirichlet Laplacian. The operator −∆ D,Γ,q has a purely discrete spectrum. In the case q = 0 identically, the latter is given by σ(−∆ D,Γ ) = λ = k 2 π 2 ℓ(e) 2 : e ∈ E, k = 1, 2, . . . , (3.2) where the multiplicity of an eigenvalue λ coincides with the number of values k and edges e for which λ = k 2 π 2 ℓ(e) 2 . In particular, where we have used the assumption that q is nonnegative, and the inclusion S ⊂ −∆ D,Γ,q implies where (·, ·) and · denote the inner product and norm respectively in L 2 (Γ). By an easy integration by parts, the adjoint of S is given by The two self-adjoint extensions of S in focus here will be the Krein-von Neumann extension of S and the Schrödinger operator with standard (also called continuity-Kirchhoff) vertex conditions.
We point out that, in general, −∆ K,Γ,q = −∆ K,Γ + q (where we interpret the latter as an additive perturbation of the Krein Laplacian); see the discussion in Remark 2.9. On the other hand, it holds that −∆ st,Γ,q = −∆ st,Γ + q, by definition.
In what follows, it will be useful to embed the study of −∆ K,Γ,q in the framework of boundary triples. The following proposition can be found in [ . . .
where v 1 , . . . , v V is an enumeration of the vertices of Γ. Then S is a closed operator and {C V , Γ 0 , Γ 1 } is a boundary triple for S * ; in particular, S has defect numbers n − = n + = V. in particular, 0 ∈ ρ(A). The value of the corresponding Weyl function at λ = 0 is

4)
where f * ∈ ker S * is arbitrary. In the potential-free case, q = 0 identically, the value of the corresponding Weyl function is where Λ q is the Dirichlet-to-Neumann matrix defined in (3.4). Moreover, In the potential-free case q = 0 identically, the domain of −∆ K,Γ consists of all

7)
where L is the weighted discrete Laplacian in (3.5).
Remark 3.5. The vertex conditions of −∆ K,Γ,q are nonlocal, i.e. they couple values of the function and its derivatives at different vertices. In the potential-free case it actually follows from (3.7) that the vertex conditions of the Krein Laplacian couple each vertex with all of its neighbours.
We calculate the vertex conditions of the Krein Laplacian explicitly for two example graphs.
Our second example shows that the Krein Laplacian and the standard Laplacian may coincide in some cases; cf. Corollary 3.9 below.
Proof. The only assertion to prove is that ran Λ q has the dimension claimed in the theorem. For the potential-free case, where Λ q = L, the weighted discrete Laplacian, it is well-known that the kernel is one-dimensional (consisting of the constant vectors), and hence its range has dimension V − 1. Now let q ≥ 0 be a nontrivial function, and let ϕ ∈ ker Λ q . Then by definition, there exists a unique f ∈ ker S * such that Γ 0 f = ϕ and Γ 1 f = 0, i.e. f ∈ ker S * ∩ dom (−∆ st,Γ,q ). In other words, f ∈ ker(−∆ st,Γ,q ). But then, by standard variational principles, Since both terms on the right-hand side are nonnegative, they are zero separately. From Γ |f ′ | 2 dx = 0, it follows that f is constant on each edge and, by continuity, constant on Γ. But then Γ q|f | 2 dx = 0 yields f = 0 identically, as q is nontrivial. Finally, ϕ = Γ 0 f = 0, so that ker Λ q = {0}. Consequently, dim ran Λ q = V , which yields the desired result.
Now the observation of Example 3.7 can be sharpened in the following way. Since flower graphs are the only graphs with V = 1, this is an immediate consequence of Theorem 3.8. In the case that q is identically zero on Γ, the latter inequality may be strengthened, Morover, one can use the inequalities for −∆ D,Γ in this case to deduce the following.
Corollary 3.10. In the case of zero potential q ≡ 0, for any λ ≥ 0, Proof. It is a straightforward exercise to show that follows from (3.2). In particular, this implies and then inserting this into (3.8) yields the desired result.
One can immediately deduce from Corollary 3.10 that the eigenvalues for −∆ K,Γ possess the Weyl asymptotics as j → ∞. However, we remark that in fact any self-adjoint extension of the operator S given by (3.1) possesses these same asymptotics.
In the following, we are going to state some eigenvalue inequalities for the perturbed Krein Laplacian. It follows directly from (3.6) that λ j+V (−∆ K,Γ,q ) = λ + j (−∆ K,Γ,q ) holds for all j ∈ N. To investigate properties of the positive eigenvalues of −∆ K,Γ,q , we first formulate the abstract variational principle in Theorem 2.4 in our specific situation.
Theorem 3.11. If Hypothesis 3.1 is satisfied, then the spectrum of −∆ K,Γ,q is purely discrete, and the positive eigenvalues for all j ∈ N. In particular, in the potential-free case q = 0 identically, holds for all j ∈ N.
The following eigenvalue inequalities and equalities are direct consequences of Theorem 2.5 and (3.6).
holds for all j ∈ N, and in the potential-free case we have for all j ∈ N. Remark 3.14. If the edge lengths in Γ are rationally independent and q ≡ 0, then the inequality (3.10) is strict for all j ∈ N, as in this case it can be seen easily that S does not possess any eigenvalues.
Remark 3.15. If Γ is a tree graph, then it is known that for the Laplacian, holds for all j ∈ N; see e.g. [43,Theorem 4.1]. One may combine this with (3.10) to obtain (3.11) in an alternative way. However, it is worth pointing out that (3.12) does not hold in general for graphs with cycles (see the discussion in [35,Section 5]), but in this case (3.11) is still true.

Spectral implications of graph surgery operations
Next, we investigate the effect of graph surgery operations on the eigenvalues of the perturbed Krein Laplacian −∆ K,Γ,q . Graph surgery refers to the process of transforming the operator by making topological changes to the metric graph, such as gluing vertices together or adding edges, forming a new graph Γ. One associates a potential q to the new graph Γ which will be determined by the type of surgery carried out. Given a surgery operation −∆ K,Γ,q → −∆ K, Γ, q , only the operators −∆ K,Γ,q and −∆ K, Γ, q will be of significance to us, and thus we use the following simplified notation for their eigenvalues throughout this section: In what follows, we always assume Hypothesis 3.1; the new potential q will satisfy the analogue of Hypothesis 3.1 conditions for Γ by construction. We begin with transformations which only affect the vertex conditions of the operator, or add new vertices. For such operations, the potential q ≡ q is unchanged (except possibly on a set of measure zero). Definition 4.1. Let Γ be the graph formed from Γ by identifying a number of its vertices, say v 1 , . . . , v k+1 , to form a new vertex v 0 . The total number of vertices is thereby reduced by k, and the potential q associated with Γ remains well-defined on Γ. The transformation −∆ K,Γ,q → −∆ K, Γ,q is called gluing vertices, and the inverse operation is referred to as cutting through vertices; cf. Figure 2.

eigenvalues (counting ground states) satisfy the interlacing inequalities
In particular, Proof. Denote by S and S the symmetric operators in L 2 (Γ) and L 2 ( Γ), respectively, defined as in (3.1). Then dom S = H 2 0 (Γ) ⊂ H 2 0 ( Γ) = dom S, and the action of the two operators coincides on the smaller domain; we always identify functions on Γ with functions on Γ, and conversely, in the obvious way. Thus S ⊂ S.
We show next that the co-dimension of dom S in dom S is k, and we do this for the case k = 1 only; for higher k this can be obtained by successively gluing vertices. For k = 1, denote by v 1 , v 2 the vertices of Γ that are glued to form the new vertex v. Let f, g ∈ dom S and observe that the linear combination h := (∂ ν g(v 1 ))f − (∂ ν f (v 1 ))g satisfies both Dirichlet and Kirchhoff conditions at both v 1 and v 2 , with the latter due to the fact that f, g satisfy Kirchhoff conditions at v (i.e. ∂ ν f (v 1 )+∂ ν f (v 2 ) = 0 and likewise for g). Then h ∈ dom S, which proves the claim on the co-dimension. Thus we can apply Theorem 2.8 to obtain inequality (4.1).
For inequality (4.2), one applies (3.6), together with the fact that the number of vertices of Γ is V − k, to the chain of inequalities (4.1) to obtain (4.2). Finally, inequality (4.3) is a trivial consequence of (3.6).
Gluing vertices therefore increases the eigenvalues of the perturbed Krein Laplacian, with inequality (4.2) providing bounds for this increase. Indeed, (4.3) implies that eigenvalues λ V −k+1 , . . . , λ V increase strictly. On the other hand, the increases are counteracted by the fact that the kernel of the operator shrinks after gluing, which explains why the positive eigenvalues actually decrease. By contrast, whilst the eigenvalues of the perturbed standard Laplacian increase by gluing, satisfying in particular the interlacing inequalities the kernel is unchanged, and thus the positive eigenvalues increase as well. . where the numbers η j are such that 0 < η j ≪ π ℓ and lim j→∞ η j = 0. Now let Γ be the loop of length ℓ, formed by gluing together the two vertices of the interval Γ; see Figure 3. According to Corollary 3.9, the Krein Laplacian −∆ K, Γ on the loop is identical to the standard Laplacian −∆ st, Γ , and thus they share the same eigenvalues. The following tables are demonstrative of Theorem 4.2 for these two graphs: the positive eigenvalues decrease by gluing, but when the ground states are included, they increase.
Definition 4.4. Assume that Hypothesis 3.1 is satisfied, and let e 0 be an edge of Γ with (possibly coincident) incident vertices v 1 , v 2 . Let Γ be the graph formed from Γ by replacing e 0 with a path graph from v 1 to v 2 , composed of two edges e 1 , e 2 , joined together by a degree-2 vertex v 0 , and with total length ℓ(e 1 ) + ℓ(e 2 ) = ℓ(e 0 ). Parametrising e 0 by [0, ℓ(e 0 )] and e 1 , e 2 by [0, ℓ(e 1 )], [ℓ(e 1 ), ℓ(e 0 )] respectively, where the endpoint ℓ(e 1 ) in both of the latter is identified with v 0 , the potential q associated with Γ is defined by on e 1 , e 2 , and q e ≡ q e on all other edges e. The transformation −∆ K,Γ,q → −∆ K, Γ, q is called inserting a degree-2 vertex along an edge, and the inverse operation is referred to as removing a degree-2 vertex ; cf.  In the special case that Γ is just one loop, it obviously does not make sense to remove the vertex of degree two, as the result would be a graph with one edge but no vertices. However, in this case the above procedure may just be understood as replacing the perturbed Krein Laplacian with the perturbed standard Laplacian. To replace "Krein vertex conditions" by standard conditions on arbitrary vertices, we refer to Theorem 6.8.
(4.4) (b) the eigenvalues (counting ground states) satisfy Proof. If we define S and S corresponding to Γ and Γ respectively, as in (3.1), then S ⊂ S. Moreover, dom S has co-dimension k 0 in dom S; indeed, if k 0 = 1, then for any two linearly independent functions f, g ∈ dom S, the function f (v 1 )g − g(v 1 )f vanishes at v 1 and thus belongs to dom S. The case of arbitrary k 0 follows inductively. Then all estimates in (4.4) follow directly from Theorem 2.8, noting that the roles of S and S are reversed. After this, (4.5) follows with the help of (3.6).
The following example shows that the positive eigenvalues of the Krein Laplacian may indeed increase strictly from adding a degree-2 vertex, in contrast with the standard Laplacian which does not feel degree-2 vertices at all. Example 4.6. Let Γ be the interval of length two, and let Γ be the path graph formed by inserting a vertex of degree 2 at its midpoint, creating two intervals each of length one connected by a single vertex. A direct computation shows that the positive eigenvalues of the Krein Laplacian on Γ are the numbers κ 2 for which κ is a root of κ (κ 2 − 2) sin(2κ) + κ + 4 sin κ − 4κ cos κ + 3κ cos(2κ) = 0.
The lowest two positive eigenvalues are then λ + 1 ≈ 4.5 2 and λ + 2 = (2π) 2 . In contrast to this, the first two positive eigenvalues of the Krein Laplacian on Γ are λ + 1 = π 2 and λ + 2 < (3π/2) 2 ; cf. Example 4.3. We have seen in Theorem 4.2 how the eigenvalues change upon gluing vertices of Γ. It is also possible to glue arbitrary points of Γ together. Again, as the (perturbed) Krein Laplacian distinguishes between vertices of degree two and nonvertex points on the graph, the following is more general than Theorem 4.2.
Definition 4.7. Assume that Hypothesis 3.1 is satisfied, and let N be a finite subset of points in Γ (which may include both vertices and points along edges). Let Γ be the graph formed by first inserting a vertex at each of the points in N which are not already vertices, and then gluing all of these new vertices together with the remaining vertices in N to form a single point. The transformation −∆ K,Γ,q → −∆ K, Γ,q is called gluing the points in N . This is evidently a two-step process, consisting of insterting degree-2 vertices along edges, and then gluing vertices. In general, one cannot determine the effect on individual eigenvalues since they increase during the first step but decrease during the second. Nevertheless, a direct application of Theorems 4.2 and 4.5 gives some insight into their behaviour.

Corollary 4.8 (Gluing arbitrary points). Assume that Hypothesis 3.1 is satisfied.
Let N be a finite subset of k + 1 points in Γ of which k 0 ≤ k + 1 are not vertices, and let Γ be the graph formed by gluing these points together. Then for the corresponding perturbed Krein Laplacians: (a) the positive eigenvalues satisfy (b) the eigenvalues (counting ground states) satisfy Next, we move on to transformations which change the volume of Γ. Here, the potential q will not be well-defined on the new graph, for which the associated potential q is defined accordingly. Definition 4.9. Assume that Hypothesis 3.1 is satisfied. Let Γ be the graph formed from Γ by lengthening one of its edges, e 0 , by a factor of α > 1, so that it has length ℓ(e 0 ) = αℓ(e 0 ) in Γ. If there is a potential q associated with Γ, then the potential q associated with Γ is defined via and q e ≡ q e on all other edges. The transformation −∆ K,Γ,q → −∆ K, Γ, q is called lengthening the edge e 0 , and the inverse operation is referred to as shrinking the edge e 0 . (4.7) (b) the eigenvalues (counting ground states) satisfy Proof. Suppose that an edge e 0 of Γ is lengthened by a factor of α > 1. Given f ∈ H 2 0 (Γ), let f be the function such that f e0 (x) = αf e0 (x/α) and f e (x) = f e (x) for all other edges e. Now, f e0 (0) = f e0 (ℓ(e 0 )) = 0, preserving the Dirichlet conditions, recalling that the potential is redefined by (4.6) on the lengthened edge. Thus Inequality (4.7) follows from Theorem 3.11, and then (4.8) from (3.6) since the kernel of the operator is unchanged by the transformation.
The remaining surgery operation deals with expanding the graph by inserting a new finite, connected metric graph Γ 0 in some way to the original graph. If there is a potential q 0 associated with Γ 0 , then we assume that it satisfies the following hypothesis in agreement with what is assumed for q on Γ.
As a rule, if no new potential is specified on the new edges, then it is reasonable to take the potential to be zero there. Nevertheless, the inequalities in Theorem 4.13 hold for the potential chosen arbitrarily there under Hypothesis 4.11. Definition 4.12. Let Hypothesis 3.1 be satisfied, and let Γ be the graph formed from Γ by gluing m of the vertices of a finite, connected metric graph Γ 0 to distinct vertices of Γ. The new potential q associated with Γ is identical to q on the edges inherited from Γ and satisfies Hypothesis 4.11 on the edges from Γ 0 . The transformation −∆ K,Γ,q → −∆ K, Γ, q is called attaching a (connected) graph to Γ (by m vertices). The inverse operation may be referred to as deleting a (connected) subgraph; cf. Figure 5.
(4.9) (b) the eigenvalues (counting ground states) satisfy here V 0 is the number of vertices of Γ 0 .
Proof. Every function in H 2 0 (Γ) can be extended by zero to a function in H 2 0 ( Γ), and this does not change the Rayleigh quotient. Thus inequality (4.9) follows from Theorem 3.11. Finally, (4.10) is obtained from (3.6), since the dimension of the kernel of the operator increases by V 0 − m.
A special case of the previous theorem consists of inserting a single edge between two vertices of Γ, a process which does not change the dimension of the kernel of the perturbed Krein Laplacian.
We emphasise that this behaviour differs substantially from the one for standard vertex conditions, where inserting an edge may either increase or decrease eigenvalues; cf. [33].

Isoperimetric inequalities
We now turn to estimates for the positive eigenvalues of the perturbed Krein Laplacian. We start with a lower estimate for the first positive eigenvalue, which we may call the spectral gap; cf. Remark 5.2 below. Proof. Let Γ be the flower graph formed from Γ by gluing all vertices. Then by Theorem 4.2 and Theorem 3.12, we have Moreover, as the only vertex of Γ has even degree (equal to twice the number of edges), we may cut through the vertex in such a way that we obtain an (Eulerian) cycle Λ of length ℓ(Γ), and by surgery principles for the perturbed standard Laplacian −∆ st, Γ,q , see e.g. [44,Theorem 4.1], we get . In the case q = 0 identically, we have hereby shown (5.1), and (5.2) follows from a direct calculation. If q is nontrivial, then I > 0, and for both operators −∆ st,Λ,q and −∆ δ,Λ,I , the smallest eigenvalue is positive. Hence we may argue further as in the proof of [26, Theorem 1]: let ψ be an eigenfunction of −∆ st,Γ,q corresponding to its lowest eigenvalue. Then , where x min is any point on Γ where |ψ| takes its minimum. Since the last quotient is the Rayleigh quotient of the Laplacian with a δ-vertex condition of strength I = Γ qdx at x min , the assertion (5.1) follows also for nontrivial potentials.
In the case of equality in (5.2), all of the above inequalities must in fact be equalities. In particular, the standard Laplacian on the flower graph Γ in the above argument already has to have 4π 2 /ℓ( Γ) 2 as its first positive eigenvalue, which is only possible if on the loop Λ resulting from splitting the central vertex of Γ, there exists an eigenfunction for the first positive eigenvalue which has the same value at each point that was glued together previously (cf. [33,Theorem 1]). Since each eigenfunction of −∆ st,Λ corresponding to the first nonzero eigenvalue takes each of its values exactly twice on the loop -at two points with distance ℓ(Γ)/2 from each other -it follows that Γ can be recovered from Λ by gluing at most two points. Hence Γ is either a loop itself or an equilateral figure-8. In other words, joining all vertices in the original graph Γ leads to a loop or a figure-8, and this is only possible if Γ is of one of the following six types: an interval, a path graph with two equal edges, a loop, an equilateral 2-cycle or an equilateral figure-8. Considering these graphs only, one finds by calculation that there exist eigenfunctions with corresponding eigenvalue 4π 2 /ℓ(Γ) 2 if and only if Γ is equilateral and has one of the four forms listed in the statement of the theorem.
Remark 5.2. The interval (0, 4π 2 /ℓ(Γ) 2 ) has empty intersection not only with the spectrum of the Krein Laplacian on one individual graph Γ. In fact, Theorem 5.1 asserts that, for fixed ℓ > 0, the interval (0, 4π 2 /ℓ 2 ) is free of spectrum for the Krein Laplacians on the whole class of metric graphs with total length ℓ. Remark 5.3. Alternatively, one may use (3.12) in combination with known lower bounds on the eigenvalues of the standard Laplacian to obtain lower bounds for the positive Krein Laplacian eigenvalues. However, using the optimal lower bound from [20], one gets which is weaker than the sharp bound (5.2).
Remark 5.4. The two crucial surgery operations used in the above proof are standard: gluing all vertices of a graph into one was used in [29], and cutting through vertices to obtain an Eulerian cycle goes back at least to [34]. Nevertheless, the above proof is slightly unusual: for the standard Laplacian, gluing vertices increases eigenvalues (the positive ones, as well as counting the ground state) whilst cutting vertices decreases them, so that both surgery operations used above -gluing all vertices into one and cutting vertices to obtain an Eulerian cycle -cannot be used within the same argument. However, in the present situation this works smoothly since gluing is performed on the positive eigenvalues of the perturbed Krein Laplacian and cutting is done only after transition to standard vertex conditions. We point out that the exact same proof also yields an estimate for higher eigenvalues in the potential-free case: Theorem 5.5. Assume that Hypothesis 3.1 is satisfied with q = 0 identically, and that Λ is a loop with the same length as for Γ. Then We conclude this section with a remark on how to apply the min-max principle to get upper spectral bounds. We do not go far into this and discuss only, very briefly, the special case of graphs which contain Eulerian cycles. We restrict ourselves here to the potential-free case, although natural generalisations for potentials exist (but their formulation may be less pleasant).
Remark 5.6. Suppose that Γ contains an Eulerian cycle Σ (obtained by cutting through vertices and removing edges not on the cycle), and let E Σ ⊆ E denote the set of edges belonging to Σ. Then the function f which on each e ∈ E Σ takes the form for some n e ∈ N, clearly satisfies Dirichlet conditions at all vertices of Σ, and, moreover, its derivatives have equal magnitude at all endpoints. Each f e contains n e /2 periods of sine, and thus, by moving around the cycle, one can ensure that Kirchhoff conditions are satisfied at all vertices of Σ by choosing appropriate signs for f e on adjacent pairs of edges; the only place where there could be a discrepancy is when one returns to the start of the cycle, as the function may end on a halfnumber of periods, but this problem is averted by imposing the further restriction that e∈EΣ n e ∈ 2N. Now, f satisfies Dirichlet-Kirchhoff conditions not only on Σ, but also on Γ, after extending it by zero on E\E Σ , so such functions provide upper estimates for the positive eigenvalues of −∆ K,Γ via the min-max principle, Theorem 3.11. The Rayleigh quotient for this f is which is an explicit upper bound for the first positive eigenvalue; the maximum value of R K [f ] among j linearly independent functions of this type gives an upper estimate for λ + j (−∆ K,Γ ). Of course, it is true in general, even with potentials, that for Γ containing an Eulerian cycle Σ, one has λ + j (−∆ K,Γ,q ) ≤ λ + j (−∆ K,Σ, q ), where q := q| Σ , due to Theorems 4.2 and 4.13.

More general perturbed Krein Laplacians
Thus far, we have studied the Krein extension of the symmetric perturbed Laplacian with Dirichlet and Kirchhoff conditions at all vertices, but the abstract theory of Krein extensions of symmetric operators allows one to extend this work to cover symmetric perturbed Laplacians with more general vertex conditions. In this section we illustrate this by considering perturbed Laplacians with "Krein vertex conditions" on a selected subset of the vertex set, and standard (continuity-Kirchhoff) vertex conditions at all further vertices. We indicate in which form the results of the previous sections carry over to this setting. The proofs are analogous in the present case and are mostly left to the reader.
Let Hypothesis 3.1 be satisfied. For B ⊂ V, define the operator S B in L 2 (Γ) by f (v) = 0 for each v ∈ B .
(6.1) Remark 6.1. A more general setting may be treated with the same methods, but we do not go into these details here: it is possible to replace the standard vertex conditions at the vertices in V \ B by any self-adjoint, local vertex conditions. For the description of such conditions, we refer the reader to [16].
Remark 6.2. The reader may think of the selected vertex set B as a kind of boundary for Γ. One choice, which may be natural in some cases, is to let B consist of all vertices of degree one. We are not restricted to this situation, but we may keep it in mind as a typical example.
The operator S B in (6.1) is symmetric, closed, and densely defined. It has defect numbers n − = n + = |B|, and is thus only self-adjoint if B = ∅. Furthermore, S B is clearly nonnegative, and its Friedrichs extension S B,F is the perturbed Krein Laplacian subject to Dirichlet boundary conditions on B and standard vertex conditions on V \ B. In particular, To derive some properties of the operator −∆ K,Γ,q,B , constructing an appropriate boundary triple is useful. Proposition 6.3. Assume that Hypothesis 3.1 is satisfied, and let B ⊂ V be nonempty. Let S B be defined in (6.1). For f ∈ dom S * B , define . . . .
where f * ∈ ker S * B is arbitrary. .
where we have written in block matrix form with respect to the decomposition of V into B and V \ B. Moreover, dim ker − ∆ K,Γ,q,B = dim ker S * B = b. Next, as an application of the abstract Theorem 2.4, we obtain the following variational characterisation for the eigenvalues of −∆ K,Γ,q,B . Theorem 6.6. If Hypothesis 3.1 is satisfied and B ⊂ V is nonempty, then the spectrum of −∆ K,Γ,q,B is purely discrete, and the positive eigenvalues λ + 1 − ∆ K,Γ,q,B ≤ λ + 2 − ∆ K,Γ,q,B ≤ . . . of −∆ K,Γ,q,B , counted with multiplicities, satisfy λ + j − ∆ K,Γ,q,B = min Analogously to Theorem 3.8, one may express the resolvent differences of −∆ K,Γ,q,B with the Friedrichs extension of S B and the perturbed standard Laplacian. In particular, one gets the following. In particular, in the potential-free case, if b = |B| = 1, then −∆ K,Γ,B equals the standard Laplacian. As a consequence of either Theorem 6.7 or Theorem 6.6, we get, analogously to (3.11), λ j+1 − ∆ st,Γ ≤ λ + j − ∆ K,Γ,B = λ j+b − ∆ K,Γ,B , j ∈ N, in the case without potential.
The surgery principles of Section 4 remain valid for the (positive) eigenvalues of the operator −∆ K,Γ,q,B , provided that all vertices involved in the surgery operations belong to B; we leave it to the reader to formulate and prove the corresponding results. Instead we formulate a related result which deals with the transition between standard and "Krein vertex conditions". Then for λ + j := λ + j − ∆ K,Γ,q,B , λ + j := λ + j − ∆ K,Γ,q, B , λ j := λ j − ∆ K,Γ,q,B , λ j := λ j − ∆ K,Γ,q, B , the following statements hold: (i) the positive eigenvalues satisfy λ + j ≤ λ + j ≤ λ + j+k ≤ λ + j+k , j ∈ N; (6.3) (ii) the eigenvalues (counting ground states) satisfy λ j ≤ λ j ≤ λ j+k ≤ λ j+k , j ∈ N. (6.4) Proof. If we denote by S and S the symmetric operators defined in (6.1) for the vertex subsets B and B respectively, then B ⊂ B implies the operator inclusion S ⊂ S. Moreover, it is easy to see that dom S has co-dimension k = b − b in dom S. Therefore (6.3) follows directly from Theorem 2.8. Using the fact that the perturbed Krein Laplacians for B and B have respectively b and b linearly independent functions in their kernels, (6.4) follows from (6.3).