The Krein-von Neumann extension for Schrödinger operators on metric graphs
aa r X i v : . [ m a t h . SP ] J un THE KREIN–VON NEUMANN EXTENSION FORSCHR ¨ODINGER OPERATORS ON METRIC GRAPHS
JACOB MULLER AND JONATHAN ROHLEDER
Dedicated with great pleasure to Henk de Snoo on the occasion of his 75th birthday
Abstract.
The Krein–von Neumann extension is studied for Schr¨odinger op-erators on metric graphs. Among other things, its vertex conditions are ex-pressed explicitly, and its relation to other self-adjoint vertex conditions (e.g.continuity-Kirchhoff) is explored. A variational characterisation for its posi-tive eigenvalues is obtained. Based on this, the behaviour of its eigenvaluesunder perturbations of the metric graph is investigated, and so-called surgeryprinciples are established. Moreover, isoperimetric eigenvalue inequalities areobtained. Introduction
It is an almost hundred-year-old story that many of the differential operatorsappearing in mathematical physics and their boundary conditions can be describedconveniently in the framework of extension theory of symmetric operators. A com-plete description of all self-adjoint extensions of a symmetric operator was firstgiven by von Neumann [40]. On the other hand, it turned out that a theory ofself-adjoint extensions of symmetric operators that are semibounded from belowcan be done conveniently by means of semibounded sesquilinear forms; this origi-nates from the work of Friedrichs [21]. However, it is due to Krein [32] (see also theworks of Vishik [49] and Birman [17]) that among all non-negative extensions of apositive definite symmetric operator S , there are two extremal ones, the Friedrichsextension S F and the (by now so-called) Krein–von Neumann extension S K , in thesense that each non-negative self-adjoint extension A of S satisfies S K ≤ A ≤ S F . These inequalities may be understood in the sense of quadratic forms or via theinvolved operators’ resolvents. It is beyond the scope of this article to provide acomplete historical review of the developments related to the Krein–von Neumannextension; for further reading we refer the reader to [2] and the survey articles[1, 8]. Among the abstract advancements on extremal extensions of positive definitesymmetric 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 do-mains, the Friedrichs extension is a very natural object; for instance, for the minimalsymmetric Laplacian on a bounded domain in R n corresponding to both Dirich-let and Neumann boundary conditions, the Friedrichs extension is the self-adjointLaplacian subject to Dirichlet boundary conditions. On the other hand, in thesame setting, the Krein–von Neumann extension corresponds to certain non-localboundary conditions which can be described in terms of the associated Dirichlet-to-Neumann map; for properties of the Krein–von Neumann extension of ellip-tic differential operators and recent related developments, we refer the reader to[4, 9, 11, 23, 24, 37]. For differential operators on metric graphs, which we consider in the presentpaper, the situation is similar, yet different in some respects. If Γ is a finite metricgraph, then we take, as a starting point, the (negative) Laplacian (i.e. the negativesecond derivative operator on each edge) S in L (Γ), which satisfies on each vertexboth Dirichlet and Kirchhoff vertex conditions; that is, the functions in the domainof S vanish and have derivatives which sum up to zero at each vertex. This sym-metric operator is very natural to carry out extension theory, since its adjoint S ∗ isthe Laplacian on Γ with continuity as its (only) vertex conditions. Therefore anyself-adjoint extension of S in L (Γ) (which, at the same time is a restriction of S ∗ )satisfies continuity conditions and thus reflects, at least to some extent, the connec-tivity of the graph. Nevertheless, the Friedrichs extension of S in this setting is theLaplacian on functions which are zero at every single vertex, an operator which,despite continuity, is determined by the graph’s edges considered as separate inter-vals, instead of the actual graph structure. Other self-adjoint extensions of S whichare more suitable for the spectral analysis of network structures are the operatorwith 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 notbeen considered much in the literature so far; an attempt for a symmetric operatorwith vertex conditions different from the ones considered here was done in [38]. TheKrein–von Neumann extension of our operator S is, like for the minimal Laplacianon 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 thevertices for functions in the domain of S K is exactly the weighted discrete Laplacianon 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 Schr¨odinger operators withnonnegative potentials q e on the edges. Namely, we consider the operator S actingas − d d x + q e on each edge e of Γ, with Dirichlet and Kirchhoff vertex conditions asdescribed 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 objectof consideration in this article. We first describe the domain of − ∆ K , Γ ,q in termsof vertex conditions and establish Krein-type formulae for the resolvent differenceswith both the Friedrichs extension (the Schr¨odinger operator with Dirichlet ver-tex conditions) and the the Schr¨odinger operator − ∆ st , Γ ,q with standard vertexconditions. As a consequence, we obtain the formuladim ran h(cid:0) − ∆ K , Γ ,q − λ (cid:1) − − (cid:0) − ∆ st , Γ ,q − λ (cid:1) − i = ( V − q = 0 identically ,V, else , in which V denotes the number of vertices of Γ and λ takes appropriate complexvalues. This formula distinguishes the potential-free case clearly from the case in-fluenced by a potential. It also sheds light on another interesting phenomenon: theKrein 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 loopsattached to it) and thus is a so-called flower graph. Moreover, we use the Krein-typeresolvent formulae to obtain some results on spectral asymptotics of the perturbedKrein Laplacian.A further property of the perturbed Krein Laplacian on a metric graph Γ, whichwe 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 isalways zero, with multiplicity equal to V , the number of vertices, as we show. All HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 3 its positive eigenvalues λ + j ( − ∆ K , Γ ,q ), ordered nondecreasingly and counted withmultiplicities, can be characterised by the variational principle λ + j (cid:0) − ∆ K , Γ ,q (cid:1) = min F ⊂ e H (Γ)dim F = j max f ∈ Ff =0 R Γ |− f ′′ + qf | d x R Γ | f ′ | d x + R Γ q | f | d x ; (1.1)here, e H (Γ) is the second-order Sobolev space on each edge, equipped with Dirich-let and Kirchhoff conditions on all edges. This formula is the exact counterpart ofa variational description of the positive eigenvalues of the perturbed Krein Lapla-cian on a domain in R n , which was established in [9, Proposition 7.5]. Before wederive (1.1), we first establish an abstract version of this principle; see Theorem2.4. Its proof is along the lines of the result for the Laplacian in [9]; however, wefound it useful and of independent interest to have it at hand also abstractly for theKrein–von Neumann extension of any symmetric, positive definite operator S forwhich 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 inequali-ties between the (positive) eigenvalues of the perturbed Krein Laplacian and otherself-adjoint extensions of S .An important field of application of the eigenvalue characterisation (1.1) areso-called surgery principles. Such principles study the influence of geometric per-turbations of a metric graph on the specta of associated Laplacians or more generaldifferential operators. The reader may think of sugery operations such as joiningtwo vertices into one or cutting through a vertex, or adding or removing edges (oreven entire subgraphs). Such principles were studied in depth for the Laplacian orSchr¨odinger 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 KreinLaplacian 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), andadding pendant edges or graphs (a process which increases the “volume” of Γ) mayonly decrease the eigenvalues. On the other hand, in some respects the behaviouris different from what we are used to for standard vertex conditions. Let us onlymention three examples: firstly, for the positive eigenvalues, gluing vertices hasactually a non-increasing effect (but at the same time also the multiplicity of theeigenvalue 0 decreases), whilst for standard vertex conditions, the positive eigen-values 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 influenceon the spectrum of an operator with standard vertex conditions. Thirdly, insertingan edge between two existing vertices makes all eigenvalues decrease (or stay thesame); 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 de-riving spectral inequalities in terms of geometric and topological parameters of thegraph such as its total length, diameter, number of edges or vertices, or its firstBetti number (or Euler characteristics, equivalently). For a few recent advances onspectral inequalities for quantum graphs, we refer to [10, 14, 28, 31, 39, 41]. Todemonstrate how surgery principles for the perturbed Krein Laplacian on a metricgraph may be applied, we establish lower bounds for the positive eigenvalues, interms of eigenvalues of a loop graph or edge lengths. For instance, for the firstpositive eigenvalue of the Krein Laplacian without potential the lower bound isexplicit, λ +1 ( − ∆ K , Γ ) ≥ (cid:18) πℓ (Γ) (cid:19) , J. MULLER AND J. ROHLEDER where ℓ (Γ) denotes the total length of Γ, and we specify the class of graphs forwhich this estimate is optimal.Considering the Krein–von Neumann (and other) extensions of a Schr¨odingeroperator 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 asymmetric Schr¨odinger operator with different vertex conditions. We mention, asan example, the Laplacian with both Dirichlet and Neumann (Kirchhoff) vertexconditions at the “loose ends”, i.e. the vertices of degree one, but standard vertexconditions at all interior vertices. In this case, the vertex conditions of the Krein–von Neumann extension will still be standard at all interior vertices, but they willcouple the vertices of degree one in a nonlocal way. We conclude our paper with ashort section where we discuss such situations.Let us briefly describe how this paper is organised. In Section 2, we reviewsome background on the abstract Krein–von Neumann extension. Moreover, weprovide a proof of the abstract counterpart of the variational principle (1.1) andderive a few easy consequences. Additionally, we study some basic properties ofboundary triples, which we use as a tool. The aim of Section 3 is to introduce theperturbed Krein Laplacian on a metric graph and to study its properties, such asa description of its domain, Krein-type resolvent formulae and some consequencesof 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 isoperimetricinequalities. Finally, Section 6 deals with the more general setting where self-adjointvertex conditions are fixed at some vertices, and extension theory is applied withrespect to the remaining vertices.2. The abstract Krein–von Neumann extension and its eigenvalues
Preliminaries.
Throughout this section we assume that H is a separablecomplex Hilbert space with inner product ( · , · ) and corresponding norm k · k . Forany closed linear operator A in H , we denote by σ ( A ) and ρ ( A ) its spectrum andresolvent set respectively. If A is self-adjoint and has a purely discrete spectrum,then we write λ ( A ) ≤ λ ( A ) ≤ . . . for its eigenvalues, counted according to their multiplicities. If G is a further Hilbertspace, we denote by B ( G , H ) the space of all bounded, everywhere-defined linearoperators from G to H and abbreviate B ( G ) := B ( G , G ).We make the following assumption. Hypothesis 2.1.
The operator S : H ⊃ dom S → H is closed and symmetric withdense domain dom S . Furthermore, S has a positive lower bound, i.e. there exists µ > Sf, f ) ≥ µ k f k , f ∈ dom S. (2.1)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 ∗ = { } andthe 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 S K f = S ∗ f, dom S K = dom S ∔ ker S ∗ . (2.2)It is well-known that S K is self-adjoint and is the smallest non-negative self-adjoint extension of S in the sense of quadratic forms. Its counterpart, the Friedrichsextension of S , is the largest non-negative extension of S and we denote it by S F . HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 5
It can be defined via completion of the quadratic form induced by S ; we do not gointo the details but refer the reader to, e.g. the discussion in [27, Chapter VI]. Forany self-adjoint, non-negative extension A of S , the relation (cid:0) ( S F − λ ) − f, f (cid:1) ≤ (cid:0) ( A − λ ) − f, f (cid:1) ≤ (cid:0) ( S K − λ ) − f, f (cid:1) , f ∈ H , holds for each λ <
0. The spectrum of the Friedrichs extension has a strictlypositive 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 thepoint 0 as the bottom of its spectrum, and the corresponding eigenspace is givenby ker S K = ker S ∗ , 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 readerto, e.g. the survey [8] for a more detailed discussion of the Krein–von Neumannextension.2.2. A variational characterisation of the positive eigenvalues of the Krein–von Neumann extension.
The main goal of this subsection is to provide an ab-stract variational description of the eigenvalues different from 0 of the Krein–vonNeumann extension. The credits for the arguments that lead to the min-max prin-ciple in Theorem 2.4 below go to the articles [7, 8, 9], where the abstract Krein–vonNeumann extension and the perturbed Krein Laplacian on domains in R n werestudied. 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 principlehere abstractly.Associated with the operator S is the space H S := dom S with norm k f k S := k Sf k , f ∈ H S . Due to (2.1), H S is a normed space, and as S is closed, it follows that H S is a Banachspace. The norm k · k S corresponds to the inner product ( f, g ) S = ( Sf, Sg ); hence H S is a Hilbert space. Moreover, there exists a constant e µ > k f k S ≥ e µ k f k , f ∈ H S . (2.3)(Indeed, if not then for each n ∈ N there exists f n ∈ H S , w.l.o.g. k f n k = 1, suchthat k Sf n k < n and hence µ ≤ ( Sf n , f n ) ≤ k Sf n k < n by (2.1), a contradictionto µ > H ∗ S the dual space of H S and write ( · , · ) H ∗ S , H S forthe sesquilinear duality between H ∗ S and H S , i.e. the continuous extension of( h, f ) H ∗ S , H S := ( h, f ) , h ∈ H , f ∈ H S , to all h ∈ H ∗ S . (Note that H is dense in H ∗ S as H S is dense in H .)It will sometimes be useful to consider S as an operator from H S to H ratherthan as an operator in H . Therefore we define e S : H S → H , e Sf := Sf, f ∈ H S . Then e S is bounded and its adjoint e S ∗ is the unique bounded operator from H to H ∗ S that satisfies (cid:0) e Sf, g (cid:1) = (cid:0) f, e S ∗ g (cid:1) H S , H ∗ S , f ∈ H S , g ∈ H . Note that on the left-hand side we might as well replace e S by S . For later use, weremark also that e S ∗ g ∈ H implies g ∈ dom S ∗ and S ∗ g = e S ∗ g . In particular,ker e S ∗ = ker S ∗ . (2.4) J. MULLER AND J. ROHLEDER
The following lemma is a variant of [7, Lemma 3.1]. For the convenience of thereader, we provide a complete proof.
Lemma 2.3.
Let Hypothesis 2.1 be satisfied. Then the operator e S ∗ S : H S → H ∗ S is bijective, and B := ( e S ∗ S ) − S : H S → H S (2.5) is a bounded, self-adjoint, non-negative operator with ker B = { } . Moreover, anumber λ > is an eigenvalue of S K if and only if λ − is an eigenvalue of B .Proof. The operator e S ∗ S is injective as e S ∗ Sf = 0 implies0 = (cid:0) e S ∗ Sf, f (cid:1) H ∗ S , H S = ( Sf, Sf ) = k Sf k , that is, f ∈ ker S which by (2.1) implies f = 0. Furthermore, let h ∈ H ∗ S . Accordingto the Fr´echet–Riesz theorem, there exists a unique f ∈ H S such that( g, h ) H S , H ∗ S = ( g, f ) S = ( Sg, Sf ) = (cid:0) g, e S ∗ Sf (cid:1) H S , H ∗ S holds for all g ∈ H S , and hence e S ∗ Sf = h . Thus e S ∗ S is bijective and, by the openmapping theorem, has a bounded inverse. In particular, the operator B in (2.5) iswell-defined and bounded as the product of two bounded operators.Let us show next that B is symmetric and thus self-adjoint. Indeed, for f ∈ H S ,we get ( Bf, f ) S = (cid:0) S ( e S ∗ S ) − Sf, Sf (cid:1) = (cid:0) e S ∗ S ( e S ∗ S ) − Sf, f (cid:1) H ∗ S , H S = ( Sf, f ) ≥ µ k f k (2.6)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 = { } .Now let λ > S K g = λg holds for some g ∈ dom S K , g = 0. Definealso f := S − 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 canwrite g = g S + g ∗ with g S ∈ dom S and g ∗ ∈ ker S ∗ and get f = S − S K g = S − Sg S + S − S ∗ g ∗ = S − S F g S = g S ∈ dom S. (2.7)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 e S ∗ Sf = λ e S ∗ g = λS ∗ ( g S + g ∗ ) = λSg S = λSf. Thus Bf = λ − f , that is, λ − is an eigenvalue of B .Conversely, let Bf = λ − f for some λ > f ∈ H S , f = 0. Then e S ∗ Sf = λSf , which can be rewritten as e S ∗ ( S − λ ) f = 0, that is, ( S − λ ) f ∈ ker e S ∗ = ker S ∗ ;see (2.4). Define g := λ − Sf . Then g is nonzero and f + λ − ( S − λ ) f = f + g − f = g, which, due to f ∈ dom S and ( S − λ ) f ∈ ker S ∗ , implies g ∈ dom S K . Finally, S K g = λ − e S ∗ Sf = Sf = λg, that is, λ is an eigenvalue of S K . This completes the proof. (cid:3) We point out that Lemma 2.3 describes, in an abstract setting, the coincidencebetween the positive eigenvalues of the Krein–von Neumann extension and theeigenvalues of an abstract buckling problem; the latter reads e S ∗ Sf = λSf and isdiscussed in detail in [7, Section 3]. HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 7
Next we provide an abstract version of the min-max principle established forKrein Laplacians on domains in [9, Proposition 7.5]. The Rayleigh quotient R K [ f ] := k Sf k ( Sf, f ) , f ∈ dom S, f = 0 , is well-defined due to (2.1). Theorem 2.4.
Assume that Hypothesis 2.1 is satisfied and that the embedding ι : H S → H is compact. Then σ ( S K ) \ { } is purely discrete, and the positiveeigenvalues λ +1 ( S K ) ≤ λ +2 ( S K ) ≤ . . . of S K , counted with multiplicities, satisfy λ + j ( S K ) = min F ⊂ dom S dim F = j max f ∈ Ff =0 R K [ f ] for all j ∈ N .Proof. As the embedding ι : H S → H is compact, it follows that the Friedrichsextension S F of S has a compact resolvent, from which it can be deduced that σ ( S K ) \ { } 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 whoseeigenvalues coincide with { λ − j : j ∈ N } . As ι is compact, the same holds for theembedding ι ∗ : H → H ∗ S , and B can be rewritten as B = ( e S ∗ S ) − ι ∗ S, which is also then compact. In particular, we can choose an orthonormal basis { f j : j ∈ N } of H S such that λ j Bf j = f j , or equivalently S ∗ Sf j = λ j Sf j , holdsfor all j ∈ N . (Here we are assuming dim H S = ∞ ; the finite-dimensional case isexactly the same with a finite orthonormal basis.) Then for each j ∈ N , R K [ f j ] = k Sf j k ( Sf j , f j ) = λ j k Sf j k ( S ∗ Sf j , f j ) = λ j holds. Let us define F := { } and F j := span { f k : k ≤ j } , j = 1 , , . . . , and denote by F ⊥ j − the orthogonal complement of F j − with respect to the innerproduct ( · , · ) S in H S for all j ∈ N . Now fix j ∈ N . Then any f ∈ F ⊥ j − can bewritten as f = P ∞ 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 thenorm in H S implies( Sf, f ) = ∞ X k = j c k ( Sf k , f ) = ∞ X k = j λ − k c k ( S ∗ Sf k , f ) = ∞ X k = j λ − k | c k | ( f k , f k ) S ≤ λ − j k f k S , and thus R K [ f ] ≥ λ j for all f ∈ F ⊥ j − , with equality for f = f j . Consequently,min f ∈ F ⊥ j − f =0 R K [ f ] = λ j . (2.8)By a similar calculation, one verifiesmax f ∈ F j f =0 R K [ f ] = λ j . (2.9) J. MULLER AND J. ROHLEDER
Now let G j ⊂ H S be a j -dimensional subspace with G j = F j . Then by a dimensionargument, there exists g j ∈ ( G j ∩ F ⊥ j − ) \ { } , and (2.8) gives λ j = min f ∈ F ⊥ j − f =0 R K [ f ] ≤ R K [ g j ] ≤ max f ∈ G j f =0 R K [ f ] . Together with (2.9), this implies the assertion of the theorem. (cid:3)
As a direct consequence, one gets the following comparison principle for thepositive 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, itfollows 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 purelydiscrete spectrum. Moreover, let d := dim ker A . Then λ j + d ( A ) ≤ λ + j ( S K ) (2.10) holds for all j ∈ N . In particular, λ j ( S F ) ≤ λ + j ( S K ) (2.11) holds for all j ∈ N . If j ∈ N is such that λ j ( S F ) is not an eigenvalue of S , then theinequality (2.11) is strict, that is, λ j ( S F ) < λ + j ( S K ) .Proof. Let us fix j and choose a j -dimensional subspace F of dom S such that k Sf k ≤ λ + j ( S K )( Sf, f ) for all f ∈ F. Then for any f ∈ F and g ∈ ker A we have (cid:0) A ( f + g ) , f + g (cid:1) ≤ k A ( f + g ) k k f + g k , and hence (cid:0) A ( f + g ) , f + g (cid:1) k f + g k ≤ k A ( f + g ) k (cid:0) A ( f + g ) , f + g (cid:1) = k Af k ( Af, f ) = k Sf k ( Sf, f ) = λ + j ( S K ) . (2.12)Due to (2.1), ker A ∩ dom S = { } 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). Notethat 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 eigenvalueof S , and let g = 0 in the estimate (2.12). Assuming λ j ( S F ) = λ + j ( S K ) for acontradiction, we get equality in (2.12), with A = S F for some nontrivial f ∈ dom S .Then f ∈ ker( S F − λ j ( S F )) ∩ dom S = ker( S − λ j ( S F )) follows, a contradiction. (cid:3) 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 discretespectrum. An example is the Krein–von Neumann extension of the Laplacian withboth Dirichlet and Neumann boundary conditions on a bounded, sufficiently smoothdomain 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 spec-trum (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 strictin general, not even if S does not have any eigenvalues.Given two symmetric operators S, e S in H such that S ⊂ e S , we get the follow-ing interlacing properties of the positive eigenvalues of their respective Krein–vonNeumann extensions. We will apply it several times in subsequent sections. HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 9
Theorem 2.8.
Let S, e S be closed, densely defined, symmetric operators in H with S ⊂ e S such that (2.1) holds for S replaced by e S . Moreover, assume that theembedding e ι : H e S → H is compact, and denote by S K and e S K the Krein–vonNeumann extensions of S and e S respectively. Then σ ( S K ) \ { } and σ ( e S K ) \ { } are purely discrete. If we assume, in addition, that dom S is a subspace of dom e S ofco-dimension k , then the positive eigenvalues of S K and e S K satisfy the interlacinginequalities λ + j ( e S K ) ≤ λ + j ( S K ) ≤ λ + j + k ( e S K ) ≤ λ + j + k ( S K ) (2.13) for all j ∈ N .Proof. Firstly, the assumption S ⊂ e S implies H S ⊂ H e S algebraically, together with k f k S = k f k e S for all f ∈ H S . Hence (2.1) follows also for S , and compactness of the embedding e ι implies com-pactness of the embedding ι : H S → H . With the help of the latter, the discretenessstatement on the spectra of S K and e S K follows from Theorem 2.4.Secondly, the first and third inclusion in (2.13) follow directly from the inclusion S ⊂ e S , and the min-max principle in Theorem 3.11. It remains to prove the middleinequality in (2.13).Let j ∈ N and let e F ⊂ dom e S be any ( j + k )-dimensional subspace of dom e S suchthat max = f ∈ e F k e Sf k ( e Sf, f ) = λ + j + k ( e S K ) . As dom S is a subspace of dom e S of co-dimension k , the subspace F := e F ∩ dom S of dom S satisfies dom F ≥ j , and we have λ + j ( S K ) ≤ max = f ∈ F k Sf k ( Sf, f ) = max = f ∈ F k e Sf k ( e Sf, f ) ≤ max = f ∈ e F k e Sf k ( e Sf, f ) = λ + j + k ( e S K ) , which completes the proof. (cid:3) We conclude this subsection with a comment on additive perturbations of theKrein–von Neumann extension.
Remark 2.9.
Assume that Q = Q ∗ is a bounded, nonnegative, everywhere definedoperator in H . If S is closed, symmetric, densely defined, and satisfies (2.1) then allthese properties are also true for S + Q , and thus S + Q has a Krein–von Neumannextension which we denote by ( S + Q ) K . It is remarkable that this operator doesnot coincide with S K + Q , the additively perturbed Krein–von Neumann extensionof 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 nontrivialkernel (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 knowthus that λ j (cid:0) ( S + Q ) K (cid:1) ≤ λ j ( S K + Q )holds for all j ∈ N . On the other hand, by our Theorem 2.5 one has λ j + d ( S K + Q ) ≤ λ + j (cid:0) ( S + Q ) K (cid:1) for all j ∈ N , where d := dim ker( S K + Q ) ≤ dim ker S K . The Krein–von Neumann extension in the framework of boundarytriples.
In this subsection, we review properties of the Krein–von Neumann ex-tension in the framework of boundary triples. Our main focus is on a Krein–typeformula that expresses the resolvent difference between the Krein–von Neumannextension 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 , Γ } consisting of aHilbert space ( G , ( · , · ) G ) and two linear mappings Γ , Γ : dom S ∗ → G is called boundary triple for S ∗ if the following conditions are satisfied:(i) the mapping { Γ , Γ } : dom S ∗ → G × G is surjective;(ii) the abstract Green identity ( S ∗ f, g ) − ( f, S ∗ g ) = (Γ f, Γ g ) G − (Γ f, Γ g ) G holds for all f, g ∈ dom S ∗ .We remark that boundary triples exist for any symmetric, densely defined op-erator S with equal defect numbers, even without the requirement (2.1). For adetailed 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 ∩ ker Γ ) = S , and twoself-adjoint extensions of S are especially distinguished, namely A := S ∗ ↾ ker Γ and B := S ∗ ↾ ker Γ . (2.14)A boundary triple comes with two operator-valued functions defined on the resol-vent set ρ ( A ) of A . Definition 2.11.
Let Hypothesis 2.1 be satisfied, and let {G , Γ , Γ } be a boundarytriple for S ∗ . The mappings γ : ρ ( A ) → B ( G , H ) and M : ρ ( A ) → B ( G )defined as γ ( λ )Γ f = f and M ( λ )Γ f = Γ f for f ∈ ker( S ∗ − λ ) are called γ -field and Weyl function respectively, associatedwith the boundary triple {G , Γ , Γ } .The well-definedness of γ ( λ ) and M ( λ ) is due to the direct sum decompositiondom S ∗ = dom A ∔ ker( S ∗ − λ ) , λ ∈ ρ ( A ) . The operator γ ( λ ) can be viewed as an abstract Poisson operator, and M ( λ ) maybe 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 caseif (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 f = ΘΓ f with a self-adjoint parameter Θ acting in G . In order to actually describe all self-adjointextensions of S , one needs to allow not only self-adjoint operators Θ but so-calledself-adjoint linear relations (or multi-valued linear operators), and we do not gointo these details here. For us it is sufficient to know the following; see e.g. [12,Theorems 2.1.3, 2.6.1, and 2.6.2]. HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 11
Proposition 2.12.
Let Hypothesis 2.1 be satisfied, let {G , Γ , Γ } be a boundarytriple for S ∗ , and let Θ be a self-adjoint operator in G . Then A Θ := S ∗ ↾ (cid:8) f ∈ dom S ∗ : Γ f = ΘΓ f (cid:9) is a self-adjoint extension of S . Moreover, if we denote by λ γ ( λ ) and λ M ( λ ) the corresponding γ -field and Weyl function respectively, and A is defined in (2.14) ,then the following assertions hold. (i) The point λ ∈ ρ ( A ) is an eigenvalue of A Θ if and only if 0 is an eigenvalueof Θ − M ( λ ) . (ii) The point λ ∈ ρ ( A ) belongs to ρ ( A Θ ) if and only if ∈ ρ (Θ − M ( λ )) . (iii) For all λ ∈ ρ ( A ) ∩ ρ ( A Θ ) , ( A Θ − λ ) − − ( A − λ ) − = γ ( λ ) (cid:0) Θ − M ( λ ) (cid:1) − γ ( λ ) ∗ holds. Characterisations analogous to item (i) in the previous theorem hold for othertypes of spectra too, such as the continuous or residual spectrum, but this is notof relevance for us in this work.If the boundary triple is chosen such that 0 ∈ ρ ( A ), then the Krein–von Neumannextension of S can be characterised in the following way; this is well-known, butfor the convenience of the reader we repeat the short proof. Proposition 2.13.
Let Hypothesis 2.1 be satisfied, and let {G , Γ , Γ } be a bound-ary triple for S ∗ such that ∈ ρ ( A ) . Moreover, let λ M ( λ ) denote the corre-sponding Weyl function. Then the Krein–von Neumann extension S K of S equals S K = S ∗ ↾ (cid:8) f ∈ dom S ∗ : Γ f = M (0)Γ f (cid:9) . Proof.
Since M (0) is self-adjoint, the restriction of S ∗ to all f which satisfy Γ f = M (0)Γ f is a self-adjoint extension of S by Proposition 2.12. Moreover, by defini-tion, each f ∈ dom S K can be written uniquely as f = f S + f ∗ with f S ∈ dom S and f ∗ ∈ ker S ∗ , and thereforeΓ f = Γ f ∗ = M (0)Γ f ∗ = M (0)Γ f, where we have used dom S = ker Γ ∩ ker Γ . This completes the proof. (cid:3) Now that we have this characterisation of the domain of S K at hand, we mayuse the above Krein–type resolvent formula to express the difference to both thedistinguished self-adjoint extensions A and B of S . Proposition 2.14.
Assume that Hypothesis 2.1 holds. Let {G , Γ , Γ } be a bound-ary triple for S ∗ , and let λ γ ( λ ) and λ M ( λ ) denote the corresponding γ -fieldand Weyl function respectively. Let A and B be given in (2.14) , and assume that ∈ ρ ( A ) . Then the following identities hold. (i) For all λ ∈ ρ ( A ) ∩ ρ ( S K ) , ( S K − λ ) − − ( A − λ ) − = γ ( λ ) (cid:0) M (0) − M ( λ ) (cid:1) − γ ( λ ) ∗ (2.15) holds. (ii) For all λ ∈ ρ ( B ) ∩ ρ ( S K ) ∩ ρ ( A ) , the operator M ( λ ) is invertible with M ( λ ) − ∈ B ( G ) and ( S K − λ ) − − ( B − λ ) − = γ ( λ ) (cid:0) M (0) − M ( λ ) (cid:1) − M (0) M ( λ ) − γ ( λ ) ∗ (2.16) holds. Proof.
Assertion (i) follows directly from plugging the result of Proposition 2.13 intothe 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( A − λ ) − − ( B − λ ) − = γ ( λ ) M ( λ ) − γ ( λ ) ∗ for all λ ∈ ρ ( A ) ∩ ρ ( B ). For those λ which additionally belong to ρ ( S K ), we combinethe latter formula with assertion (i) of the present proposition to get( S K − λ ) − − ( B − λ ) − = γ ( λ ) h(cid:0) M (0) − M ( λ ) (cid:1) − + M ( λ ) − i γ ( λ ) ∗ . From this, the assertion (ii) follows by an easy calculation left to the reader. (cid:3)
The resolvent formulae in the previous proposition may be used to determinethe rank of the resolvent differences as follows.
Corollary 2.15.
Assume that Hypothesis 2.1 is satisfied. Let {G , Γ , Γ } be aboundary triple for S ∗ with Weyl function λ M ( λ ) , and let A, B be as definedin (2.14) . Moreover, let ∈ ρ ( A ) . Then the following hold. (i) For all λ ∈ ρ ( A ) ∩ ρ ( S K ) , dim ran (cid:2) ( S K − λ ) − − ( A − λ ) − (cid:3) = dim ker( S ∗ − λ ) = dim G . (ii) For all λ ∈ ρ ( A ) ∩ ρ ( B ) ∩ ρ ( S K ) , dim ran (cid:2) ( S K − λ ) − − ( B − λ ) − (cid:3) = dim ran M (0) . Proof.
This follows rather directly from formulas (2.15) and (2.16) in a way similarto the proof of [12, Theorem 2.8.3]. In fact, we use that γ ( λ ) : G → ker( S ∗ − λ ) isan isomorphism and thatker γ ( λ ) ∗ = (cid:0) ran γ ( λ ) (cid:1) ⊥ = (cid:0) ker( S ∗ − λ ) (cid:1) ⊥ . This implies thatran (cid:2) ( S K − λ ) − − ( A − λ ) − (cid:3) = ran (cid:2) ( S K − λ ) − − ( A − λ ) − (cid:3) ↾ ker( S ∗ − λ ) , for all λ ∈ ρ ( A ) ∩ ρ ( S K ), with the same equation holding after replacing A with B for all λ ∈ ρ ( A ) ∩ ρ ( B ) ∩ ρ ( S K ). Finally, as γ ( λ ) ∗ ↾ ker( S ∗ − λ ) : ker( S ∗ − λ ) → G is an isomorphism and both ( M (0) − M ( λ )) − and M ( λ ) − are isomorphisms of G ,the desired result follows from (2.15) and (2.16). (cid:3) Perturbed Krein Laplacians on metric graphs
In this section and all sections which follow, we assume that Γ is a metric graphconsisting 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 willalways assume that Γ is finite, i.e. V := |V| and E := |E| are finite numbers, andwe 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 (Γ) if f e ∈ L (0 , ℓ ( e )) for each e ∈ E . In order to define Schr¨odinger operators on metric graphs we make use ofthe Sobolev spaces e H k (Γ) := (cid:8) f ∈ L (Γ) : f e ∈ H k (0 , ℓ ( e )) for each e ∈ E (cid:9) ,k ∈ N . For functions in e H (Γ), we may talk about continuity at a vertex v , meaningthat for any two edges e, ˆ e incident with v , the limit values (or traces) of f e and f ˆ e HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 13 at the endpoints of the edges corresponding to v coincide. In this sense, we makeuse of the function space H (Γ) := n f ∈ e H (Γ) : f is continuous at each vertex o . Moreover, for f ∈ e H (Γ) and v ∈ V , we write ∂ ν f ( v ) := X ∂f e ( v ) , where the sum is taken over all edges e incident with v , and ∂f e ( v ) is the derivativeof 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 Schr¨odinger operators on metric graphs with potentials thatare, for simplicity, bounded. However, everything may be extended easily to form-bounded (i.e. L ) potentials. We will always assume the following hypothesis. Hypothesis 3.1.
On the finite, connected metric graph Γ, the potential q : Γ → R is measurable and bounded, and q ( x ) ≥ x ∈ Γ.Under Hypothesis 3.1, we define the Schr¨odinger operator with potential q sub-ject to Dirichlet and Kirchhoff vertex conditions at all vertices,( Sf ) e = − f ′′ e + q e f e on each edge e ∈ E , dom S = e H (Γ) := n f ∈ e H (Γ) ∩ H (Γ) : ∂ ν f ( v ) = f ( v ) = 0 for each v ∈ V o . (3.1)It is easy to see that S is a symmetric, nonnegative, densely defined operator in theHilbert space L (Γ). Since ⊕ e ∈E C ∞ (0 , ℓ ( e )) ⊂ dom S , the Friedrichs extension of S is the operator − ∆ D , Γ ,q , called the perturbed Dirichlet Laplacian , given by( − ∆ D , Γ ,q f ) e = − f ′′ e + q e f e on each edge e ∈ E , dom ( − ∆ D , Γ ,q ) = n f ∈ e H (Γ) ∩ H (Γ) : f ( v ) = 0 for each v ∈ V o ;if q = 0 identically, we just write − ∆ D , Γ and call it the Dirichlet Laplacian . Theoperator − ∆ D , Γ ,q has a purely discrete spectrum. In the case q = 0 identically, thelatter is given by σ ( − ∆ D , Γ ) = (cid:26) λ = k π ℓ ( e ) : e ∈ E , k = 1 , , . . . (cid:27) , (3.2)where the multiplicity of an eigenvalue λ coincides with the number of values k andedges e for which λ = k π ℓ ( e ) . In particular,min σ ( − ∆ D , Γ ,q ) ≥ min σ ( − ∆ D , Γ ) = π (max e ∈E ℓ ( e )) =: µ > , where we have used the assumption that q is nonnegative, and the inclusion S ⊂− ∆ D , Γ ,q implies ( Sf, f ) ≥ µ k f k , f ∈ dom S, where ( · , · ) and k · k denote the inner product and norm respectively in L (Γ). Byan easy integration by parts, the adjoint of S is given by( S ∗ f ) e = − f ′′ e + q e f e on each edge e ∈ E , dom S ∗ = e H (Γ) ∩ H (Γ) . The two self-adjoint extensions of S in focus here will be the Krein–von Neumannextension of S and the Schr¨odinger operator with standard (also called continuity-Kirchhoff) vertex conditions. Definition 3.2.
We assume that Hypothesis 3.1 is satisfied.(i) The perturbed Krein Laplacian on Γ is the Krein–von Neumann extension − ∆ K , Γ ,q := S K of S .(ii) The perturbed standard Laplacian on Γ is the operator given by( − ∆ st , Γ ,q f ) e = − f ′′ e + q e f e on each edge e ∈ E , dom ( − ∆ st , Γ ,q ) = n f ∈ e H (Γ) ∩ H (Γ) : ∂ ν f ( v ) = 0 for each v ∈ V o . The corresponding vertex conditions are called standard conditions .In the case that the potenial q is identically zero, we write − ∆ K , Γ := − ∆ K , Γ , and − ∆ st , Γ := − ∆ st , Γ , and call these operators Krein Laplacian and standardLaplacian , respectively.We point out that, in general, − ∆ K , Γ ,q = − ∆ K , Γ + q (where we interpret thelatter as an additive perturbation of the Krein Laplacian); see the discussion inRemark 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 frameworkof boundary triples. The following proposition can be found in [19, Lemma 2.14 andTheorem 2.16]; see also [13, Proposition 10.1]. For the statement on the weighteddiscrete Laplacian, see e.g. Step 2 in the proof of [22, Proposition 3.1]. Proposition 3.3.
Assume that Hypothesis 3.1 is satisfied, and let S be definedin (3.1) . For f ∈ dom S ∗ = e H (Γ) ∩ H (Γ) , define Γ f = f ( v ) ... f ( v V ) and Γ f = − ∂ ν f ( v ) ... − ∂ ν f ( v V ) , where v , . . . , v V is an enumeration of the vertices of Γ . Then S is a closed operatorand { C V , Γ , Γ } is a boundary triple for S ∗ ; in particular, S has defect numbers n − = n + = V. (3.3) The corresponding extensions A and B of S defined in (2.14) are given by A = − ∆ D , Γ ,q and B = − ∆ st , Γ ,q ; in particular, ∈ ρ ( A ) . The value of the corresponding Weyl function at λ = 0 is M (0) = − Λ q , where Λ q is the Dirichlet-to-Neumann matrix defined via the relation ∂ ν f ∗ ( v ) ... ∂ ν f ∗ ( v V ) = Λ q f ∗ ( v ) ... f ∗ ( v V ) , (3.4) where f ∗ ∈ ker S ∗ is arbitrary. In the potential-free case, q = 0 identically, thevalue of the corresponding Weyl function is M (0) = − Λ = − L , where L is theweighted discrete Laplacian L defined as L i,j = − P e connects v i and v j L ( e ) if v i and v j are adjacent , i = j, if v i , v j are not adjacent , P e ∈E ( v i ) , e no loop L ( e ) if i = j. (3.5)This proposition allows us to describe the domain of − ∆ K , Γ ,q in terms of itsvertex conditions and to obtain some properties of the perturbed Krein Laplacianright away. The next proposition follows immediately from Proposition 3.3 togetherwith Proposition 2.13. Furthermore, from (3.3) we obtain the multiplicity of thezero eigenvalue of − ∆ K , Γ ,q . HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 15
Proposition 3.4.
Under Hypothesis 3.1 the perturbed Krein Laplacian acts as (cid:0) − ∆ K , Γ ,q f (cid:1) e = − f ′′ e + q e f e on each edge e ∈ E , and its domain consists of all f ∈ e H (Γ) ∩ H (Γ) such that ∂ ν f ( v ) ... ∂ ν f ( v V ) = Λ q f ( v ) ... f ( v V ) , where Λ q is the Dirichlet-to-Neumann matrix defined in (3.4) . Moreover, dim ker (cid:0) − ∆ K , Γ ,q (cid:1) = dim ker S ∗ = V. (3.6) In the potential-free case q = 0 identically, the domain of − ∆ K , Γ consists of all f ∈ e H (Γ) ∩ H (Γ) which satisfy the vertex conditions ∂ ν f ( v ) ... ∂ ν f ( v V ) = L f ( v ) ... f ( v V ) , (3.7) 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 valuesof the function and its derivatives at different vertices. In the potential-free case itactually follows from (3.7) that the vertex conditions of the Krein Laplacian coupleeach vertex with all of its neighbours.We calculate the vertex conditions of the Krein Laplacian explicitly for twoexample graphs. Example 3.6.
Let Γ = [0 , ℓ ] be an interval, i.e. a graph consisting of two verticesand one edge between them. On this graph, the weighted discrete Laplacian L defined in (3.5) equals L = 1 ℓ (cid:18) − − (cid:19) and the vertex condition for the Krein Laplacian − ∆ K , Γ as described in Proposi-tion 3.4 can be rewritten f ′ (0) = f ′ ( ℓ ) , f ( ℓ ) = f (0) + ℓf ′ (0) . Our second example shows that the Krein Laplacian and the standard Laplacianmay coincide in some cases; cf. Corollary 3.9 below.
Example 3.7.
Let Γ be a flower graph, i.e. a graph with one vertex and E loopsattached to it; special cases are loops ( E = 1) and figure-8 graphs ( E = 2); cf.Figure 1. Then any function f ∗ which is harmonic on every edge and belongsto H (Γ) is necessarily constant on all of Γ. Thus each f ∈ dom ( − ∆ K , Γ ) satisfies f = f S + c with f S ∈ dom S and c constant; in particular, f ∈ dom ( − ∆ st , Γ ), thedomain of the standard Laplacian on Γ. As both − ∆ K , Γ and − ∆ st , Γ are self-adjointoperators, they coincide, − ∆ K , Γ = − ∆ st , Γ , on any flower graph Γ.Next we compare the perturbed Krein Laplacian with the perturbed DirichletLaplacian and the perturbed standard Laplacian. We apply Proposition 3.4 andCorollary 2.15 to the boundary triple in Proposition 3.3 and get the following result. Theorem 3.8.
Assume that Hypothesis 3.1 is satisfied. Let λ γ ( λ ) and λ M ( λ ) be the γ -field and Weyl function respectively corresponding to the boundarytriple in Proposition 3.3. Figure 1.
A “generic” flower graph and two special cases, theloop and the figure-8 graph.(i)
For λ ∈ ρ ( − ∆ K , Γ ,q ) ∩ ρ ( − ∆ D , Γ ,q ) , the formula (cid:0) − ∆ K , Γ ,q − λ (cid:1) − − (cid:0) − ∆ D , Γ ,q − λ (cid:1) − = − γ ( λ ) (cid:0) Λ q + M ( λ ) (cid:1) − γ ( λ ) ∗ holds. In particular, dim ran h(cid:0) − ∆ K , Γ ,q − λ (cid:1) − − (cid:0) − ∆ D , Γ ,q − λ (cid:1) − i = V. (ii) For λ ∈ ρ ( − ∆ K , Γ ,q ) ∩ ρ ( − ∆ st , Γ ,q ) ∩ ρ ( − ∆ D , Γ ,q ) , the formula (cid:0) − ∆ K , Γ ,q − λ (cid:1) − − (cid:0) − ∆ st , Γ ,q − λ (cid:1) − = γ ( λ ) (cid:0) Λ q + M ( λ ) (cid:1) − Λ q M ( λ ) − γ ( λ ) ∗ holds. In particular, dim ran h(cid:0) − ∆ K , Γ ,q − λ (cid:1) − − (cid:0) − ∆ st , Γ ,q − λ (cid:1) − i = dim ran Λ q = ( V − if q = 0 identically ,V, else . Proof.
The only assertion to prove is that ran Λ q has the dimension claimed inthe theorem. For the potential-free case, where Λ q = L , the weighted discreteLaplacian, it is well-known that the kernel is one-dimensional (consisting of theconstant vectors), and hence its range has dimension V −
1. Now let q ≥ ϕ ∈ ker Λ q . Then by definition, there exists a unique f ∈ ker S ∗ such that Γ f = ϕ and Γ f = 0, i.e. f ∈ ker S ∗ ∩ dom ( − ∆ st , Γ ,q ). Inother words, f ∈ ker( − ∆ st , Γ ,q ). But then, by standard variational principles,0 = Z Γ | f ′ | d x + Z Γ q | f | d x. Since both terms on the right-hand side are nonnegative, they are zero separately.From R Γ | f ′ | d x = 0, it follows that f is constant on each edge and, by continuity,constant on Γ. But then R Γ q | f | d x = 0 yields f = 0 identically, as q is nontrivial.Finally, ϕ = Γ f = 0, so that ker Λ q = { } . Consequently, dim ran Λ q = V , whichyields the desired result. (cid:3) Now the observation of Example 3.7 can be sharpened in the following way. Sinceflower graphs are the only graphs with V = 1, this is an immediate consequence ofTheorem 3.8. Corollary 3.9.
Under Hypothesis 3.1, the following statements are equivalent. (i)
The perturbed Krein Laplacian and the perturbed standard Laplacian coin-cide, i.e. − ∆ K , Γ ,q = − ∆ st , Γ ,q ; (ii) Γ is a flower graph and q = 0 identically. HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 17
Theorem 3.8 allows us to deduce eigenvalue asymptotics for the perturbed KreinLaplacian: for λ ≥ N (cid:0) λ ; − ∆ • , Γ ,q (cid:1) := (cid:12)(cid:12) σ (cid:0) − ∆ • , Γ ,q (cid:1) ∩ ( −∞ , λ ] (cid:12)(cid:12) , • = K , D , st , the number of eigenvalues of the respective operator up to λ . Under Hypothesis 3.1,it follows immediately from Theorem 3.8 and the minimality property of the Krein–von Neumann extension that N (cid:0) λ ; − ∆ D , Γ ,q (cid:1) ≤ N (cid:0) λ ; − ∆ K , Γ ,q (cid:1) ≤ N (cid:0) λ ; − ∆ D , Γ ,q (cid:1) + V, (3.8) N (cid:0) λ ; − ∆ st , Γ ,q (cid:1) ≤ N (cid:0) λ ; − ∆ K , Γ ,q (cid:1) ≤ N (cid:0) λ ; − ∆ st , Γ ,q (cid:1) + V. In the case that q is identically zero on Γ, the latter inequality may be strengthened, N (cid:0) λ ; − ∆ st , Γ (cid:1) ≤ N (cid:0) λ ; − ∆ K , Γ (cid:1) ≤ N (cid:0) λ ; − ∆ st , Γ (cid:1) + V − . (3.9)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 ≡ , for any λ ≥ , ℓ (Γ) π √ λ − E ≤ N (cid:0) λ ; − ∆ K , Γ (cid:1) ≤ ℓ (Γ) π √ λ + V. Proof.
It is a straightforward exercise to show that N ( λ ; − ∆ D , Γ ) = X e ∈E (cid:22) ℓ ( e ) π √ λ (cid:23) follows from (3.2). In particular, this implies ℓ (Γ) π √ λ − E ≤ N ( λ ; − ∆ D , Γ ) ≤ ℓ (Γ) π √ λ, and then inserting this into (3.8) yields the desired result. (cid:3) One can immediately deduce from Corollary 3.10 that the eigenvalues for − ∆ K , Γ possess the Weyl asymptotics λ j ( − ∆ K , Γ ) ∼ (cid:18) jπℓ (Γ) (cid:19) as j → ∞ . However, we remark that in fact any self-adjoint extension of theoperator S given by (3.1) possesses these same asymptotics.In the following, we are going to state some eigenvalue inequalities for the per-turbed 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 specificsituation. Theorem 3.11.
If Hypothesis 3.1 is satisfied, then the spectrum of − ∆ K , Γ ,q ispurely discrete, and the positive eigenvalues λ +1 (cid:0) − ∆ K , Γ ,q (cid:1) ≤ λ +2 (cid:0) − ∆ K , Γ ,q (cid:1) ≤ . . . of − ∆ K , Γ ,q , counted with multiplicities, satisfy λ + j (cid:0) − ∆ K , Γ ,q (cid:1) = min F ⊂ e H (Γ)dim F = j max f ∈ Ff =0 R Γ |− f ′′ + qf | d x R Γ | f ′ | d x + R Γ q | f | d x for all j ∈ N . In particular, in the potential-free case q = 0 identically, λ + j ( − ∆ K , Γ ) = min F ⊂ e H (Γ)dim F = j max f ∈ Ff =0 R Γ | f ′′ | d x R Γ | f ′ | d x holds for all j ∈ N . The following eigenvalue inequalities and equalities are direct consequences ofTheorem 2.5 and (3.6).
Theorem 3.12.
Let Hypothesis 3.1 be satisfied, let − ∆ Γ ,q be any self-adjoint ex-tension of the operator S in (3.1) , and let d := dim ker( − ∆ Γ ,q ) . Then λ j + d ( − ∆ Γ ,q ) ≤ λ + j ( − ∆ K , Γ ,q ) = λ j + V ( − ∆ K , Γ ,q ) holds for all j ∈ N . In particular, λ j ( − ∆ D , Γ ,q ) ≤ λ + j ( − ∆ K , Γ ,q ) = λ j + V ( − ∆ K , Γ ,q ) (3.10) holds for all j ∈ N , and in the potential-free case we have λ j +1 ( − ∆ st , Γ ) ≤ λ + j ( − ∆ K , Γ ) = λ j + V ( − ∆ K , Γ ) (3.11) for all j ∈ N . Remark 3.13.
The inequalities (3.10) and (3.11) can alternatively be deducedfrom Theorem 3.8; cf. (3.8) and (3.9).
Remark 3.14.
If the edge lengths in Γ are rationally independent and q ≡
0, thenthe 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, λ j +1 ( − ∆ st , Γ ) ≤ λ j ( − ∆ D , Γ ) (3.12)holds for all j ∈ N ; see e.g. [43, Theorem 4.1]. One may combine this with (3.10) toobtain (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, Section5]), but in this case (3.11) is still true.4. Spectral implications of graph surgery operations
Next, we investigate the effect of graph surgery operations on the eigenvaluesof the perturbed Krein Laplacian − ∆ K , Γ ,q . Graph surgery refers to the process oftransforming the operator by making topological changes to the metric graph, suchas gluing vertices together or adding edges, forming a new graph e Γ. One associatesa potential e q to the new graph e Γ which will be determined by the type of surgerycarried out. Given a surgery operation − ∆ K , Γ ,q
7→ − ∆ K , e Γ , e q , only the operators − ∆ K , Γ ,q and − ∆ K , e Γ , e q will be of significance to us, and thus we use the followingsimplified notation for their eigenvalues throughout this section: λ + j := λ + j ( − ∆ K , Γ ,q ) , e λ + j := λ + j ( − ∆ K , e Γ , e q ) ,λ j := λ j ( − ∆ K , Γ ,q ) , e λ j := λ j ( − ∆ K , e Γ , e q ) . In what follows, we always assume Hypothesis 3.1; the new potential e q will satisfythe analogue of Hypothesis 3.1 conditions for e Γ by construction.We begin with transformations which only affect the vertex conditions of theoperator, or add new vertices. For such operations, the potential e q ≡ q is unchanged(except possibly on a set of measure zero). HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 19
Definition 4.1.
Let e Γ be the graph formed from Γ by identifying a number of itsvertices, say v , . . . , v k +1 , to form a new vertex v . The total number of vertices isthereby reduced by k , and the potential q associated with Γ remains well-definedon e Γ. The transformation − ∆ K , Γ ,q
7→ − ∆ K , e Γ ,q is called gluing vertices , and theinverse operation is referred to as cutting through vertices ; cf. Figure 2. • • • • •• • • v v v • • •• • • v Figure 2.
Gluing vertices v , v , v of Γ to form a new vertex v of e Γ. Theorem 4.2 (Gluing vertices) . Let Hypothesis 3.1 be satisfied, and let e Γ be thegraph formed by gluing precisely k + 1 vertices of Γ . Then for the correspondingperturbed Krein Laplacians: (a) the positive eigenvalues satisfy the interlacing inequalities e λ + j ≤ λ + j ≤ e λ + j + k ≤ λ + j + k , j ∈ N ; (4.1)(b) the eigenvalues (counting ground states) satisfy the interlacing inequalities λ j ≤ e λ j ≤ λ j + k ≤ e λ j + k , j ∈ N . (4.2) In particular, λ V < e λ V − k +1 . (4.3) Proof.
Denote by S and e S the symmetric operators in L (Γ) and L ( e Γ), respec-tively, defined as in (3.1). Thendom S = e H (Γ) ⊂ e H ( e Γ) = dom e S, and the action of the two operators coincides on the smaller domain; we alwaysidentify functions on Γ with functions on e Γ, and conversely, in the obvious way.Thus S ⊂ e S .We show next that the co-dimension of dom S in dom e S is k , and we do thisfor the case k = 1 only; for higher k this can be obtained by successively gluingvertices. For k = 1, denote by v , v the vertices of Γ that are glued to form thenew vertex v . Let f, g ∈ dom e S and observe that the linear combination h := ( ∂ ν g ( v )) f − ( ∂ ν f ( v )) g satisfies both Dirichlet and Kirchhoff conditions at both v and v , with the latterdue to the fact that f, g satisfy Kirchhoff conditions at v (i.e. ∂ ν f ( v )+ ∂ ν f ( v ) = 0and 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 numberof vertices of e 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). (cid:3) Gluing vertices therefore increases the eigenvalues of the perturbed Krein Lapla-cian, with inequality (4.2) providing bounds for this increase. Indeed, (4.3) impliesthat eigenvalues λ V − k +1 , . . . , λ V increase strictly. On the other hand, the increasesare counteracted by the fact that the kernel of the operator shrinks after gluing,which explains why the positive eigenvalues actually decrease. By contrast, whilstthe eigenvalues of the perturbed standard Laplacian increase by gluing, satisfyingin particular the interlacing inequalities λ j ( − ∆ st , Γ ,q ) ≤ λ j ( − ∆ st , e Γ ,q ) ≤ λ j + k ( − ∆ st , Γ ,q ) ≤ λ j + k ( − ∆ st , e Γ ,q ) , the kernel is unchanged, and thus the positive eigenvalues increase as well. Example 4.3.
Let Γ = [0 , ℓ ] be the interval of length ℓ . The vertex conditionsfor the Krein Laplacian − ∆ K , Γ were calculated in Example 3.6. From this, onecomputes that the eigenvalues λ = κ are given by the solutions of the equation (cid:20) cos κℓ − κℓ sin κℓ (cid:21) sin κℓ . The positive solutions to this are κ = ( jπℓ if j = 2 , , , ... jπℓ − η j if j = 3 , , , ... where the numbers η j are such that 0 < η j ≪ πℓ and lim j →∞ η j = 0.Now let e Γ be the loop of length ℓ , formed by gluing together the two verticesof the interval Γ; see Figure 3. According to Corollary 3.9, the Krein Laplacian − ∆ K , e Γ on the loop is identical to the standard Laplacian − ∆ st , e Γ , and thus theyshare the same eigenvalues. • • ℓ • ℓ Figure 3.
Transforming the interval Γ to the loop e Γ.The following tables are demonstrative of Theorem 4.2 for these two graphs: thepositive eigenvalues decrease by gluing, but when the ground states are included,they increase.
Table 1.
Positive eigenvalues j λ + j ( − ∆ K , Γ ) λ + j ( − ∆ K , e Γ )1 (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ − η (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ − η (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ − η (cid:1) (cid:0) πℓ (cid:1) Table 2.
All eigenvalues j λ j ( − ∆ K , Γ ) λ j ( − ∆ K , e Γ )1 0 02 0 (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ − η (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ (cid:1) (cid:0) πℓ − η (cid:1) (cid:0) πℓ (cid:1) Definition 4.4.
Assume that Hypothesis 3.1 is satisfied, and let e be an edge ofΓ with (possibly coincident) incident vertices v , v . Let e Γ be the graph formedfrom Γ by replacing e with a path graph from v to v , composed of two edges e , e , joined together by a degree-2 vertex v , and with total length ℓ ( e ) + ℓ ( e ) = ℓ ( e ). Parametrising e by [0 , ℓ ( e )] and e , e by [0 , ℓ ( e )] , [ ℓ ( e ) , ℓ ( e )] respectively,where the endpoint ℓ ( e ) in both of the latter is identified with v , the potential e q associated with e Γ is defined by e q e := q e | [0 ,ℓ ( e )] , e q e := q e | [ ℓ ( e ) ,ℓ ( e )] on e , e , and e q e ≡ q e on all other edges e . The transformation − ∆ K , Γ ,q
7→ − ∆ K , e Γ , e q is called inserting a degree-2 vertex along an edge , and the inverse operation isreferred to as removing a degree-2 vertex ; cf. Figure 4. • • • v e e ℓ ( e ) ℓ ( e ) ℓ ( e ) • • e ℓ ( e ) Figure 4.
Inserting a degree-2 vertex v of Γ to form e Γ.In the special case that Γ is just one loop, it obviously does not make sense toremove the vertex of degree two, as the result would be a graph with one edge butno vertices. However, in this case the above procedure may just be understood asreplacing 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.
Theorem 4.5 (Inserting degree-2 vertices) . Let Hypothesis 3.1 be satisfied, and let e Γ be the graph formed by inserting k vertices of degree 2 along edges of Γ . Thenfor the corresponding Krein Laplacians: (a) the positive eigenvalues satisfy λ + j ≤ e λ + j ≤ λ + j + k ≤ e λ + j + k , j ∈ N ; (4.4)(b) the eigenvalues (counting ground states) satisfy e λ j ≤ λ j ≤ e λ j + k ≤ λ j + k , j ∈ N . (4.5) Proof.
If we define S and e S corresponding to Γ and e Γ respectively, as in (3.1),then e S ⊂ S . Moreover, dom e S has co-dimension k in dom S ; indeed, if k = 1,then for any two linearly independent functions f, g ∈ dom S , the function f ( v ) g − g ( v ) f vanishes at v and thus belongs to dom e S . The case of arbitrary k followsinductively. Then all estimates in (4.4) follow directly from Theorem 2.8, notingthat the roles of S and e S are reversed. After this, (4.5) follows with the help of(3.6). (cid:3) The following example shows that the positive eigenvalues of the Krein Laplacianmay indeed increase strictly from adding a degree-2 vertex, in contrast with thestandard Laplacian which does not feel degree-2 vertices at all.
Example 4.6.
Let Γ be the interval of length two, and let e Γ be the path graphformed by inserting a vertex of degree 2 at its midpoint, creating two intervals eachof length one connected by a single vertex. A direct computation shows that the positive eigenvalues of the Krein Laplacian on e Γ are the numbers κ for which κ isa root of κ (cid:0) ( κ −
2) sin(2 κ ) + κ + 4 sin κ − κ cos κ + 3 κ cos(2 κ ) (cid:1) = 0 . The lowest two positive eigenvalues are then e λ +1 ≈ . and e λ +2 = (2 π ) . In contrastto this, the first two positive eigenvalues of the Krein Laplacian on Γ are λ +1 = π and λ +2 < (3 π/ ; cf. Example 4.3.We have seen in Theorem 4.2 how the eigenvalues change upon gluing verticesof 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 non-vertex 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 finitesubset of points in Γ (which may include both vertices and points along edges). Let e Γ be the graph formed by first inserting a vertex at each of the points in N whichare not already vertices, and then gluing all of these new vertices together with theremaining vertices in N to form a single point. The transformation − ∆ K , Γ ,q ∆ K , e Γ ,q is called gluing the points in N .This is evidently a two-step process, consisting of insterting degree-2 verticesalong edges, and then gluing vertices. In general, one cannot determine the effecton individual eigenvalues since they increase during the first step but decreaseduring the second. Nevertheless, a direct application of Theorems 4.2 and 4.5 givessome 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 ≤ k + 1 are not vertices, andlet e Γ be the graph formed by gluing these points together. Then for the correspondingperturbed Krein Laplacians: (a) the positive eigenvalues satisfy e λ + j ≤ λ + j + k ≤ e λ + j + k + k ≤ λ + j + k +2 k , j ∈ N ;(b) the eigenvalues (counting ground states) satisfy e λ j ≤ λ j + k ≤ e λ j + k + k ≤ λ j +2 k + k , j ∈ N . Next, we move on to transformations which change the volume of Γ. Here, thepotential q will not be well-defined on the new graph, for which the associatedpotential e q is defined accordingly. Definition 4.9.
Assume that Hypothesis 3.1 is satisfied. Let e Γ be the graph formedfrom Γ by lengthening one of its edges, e , by a factor of α >
1, so that it has length e ℓ ( e ) = αℓ ( e ) in e Γ. If there is a potential q associated with Γ, then the potential e q associated with e Γ is defined via e q e ( x ) := α − q e ( x/α ) , (4.6)and e q e ≡ q e on all other edges. The transformation − ∆ K , Γ ,q
7→ − ∆ K , e Γ , e q is called lengthening the edge e , and the inverse operation is referred to as shrinking theedge e . Theorem 4.10 (Lengthening an edge) . Let Hypothesis 3.1 be satisfied, and let e Γ bethe graph formed by lengthening one of the edges of Γ . Then for the correspondingperturbed Krein Laplacians: (a) the positive eigenvalues satisfy e λ + j ≤ λ + j , j ∈ N ; (4.7) HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 23 (b) the eigenvalues (counting ground states) satisfy e λ j ≤ λ j , j ∈ N . (4.8) Proof.
Suppose that an edge e of Γ is lengthened by a factor of α >
1. Given f ∈ e H (Γ), let e f be the function such that e f e ( x ) = αf e ( x/α ) and e f e ( x ) = f e ( x ) forall other edges e . Now, e f e (0) = e f e ( ℓ ( e )) = 0, preserving the Dirichlet conditions,and e f ′ e (0) = f ′ e (0), e f ′ e ( ℓ ( e )) = f ′ e ( ℓ ( e )), preserving the Kirchhoff conditions,whence e f ∈ e H ( e Γ). Notice that e H (Γ) → e H ( e Γ) : f e f is a bijection. Then Z e ℓ ( e )0 | e f ′ e | d x + Z e ℓ ( e )0 e q e | e f e | d x = α Z ℓ ( e )0 | f ′ e | d x + α Z ℓ ( e )0 q e | f e | d x, and Z e ℓ ( e )0 (cid:12)(cid:12)(cid:12) − e f ′′ e + e q e e f e (cid:12)(cid:12)(cid:12) d x = 1 α Z ℓ ( e )0 (cid:12)(cid:12) − f ′′ e + q e f e (cid:12)(cid:12) d x, recalling that the potential is redefined by (4.6) on the lengthened edge. Thus R e Γ | − e f ′′ + e q e f | d x R e Γ | e f ′ | d x + R e Γ e q | e f | d x ≤ R Γ | − f ′ + qf ′ | d x R Γ | f ′ | d x + R Γ q | f | d x . Inequality (4.7) follows from Theorem 3.11, and then (4.8) from (3.6) since thekernel of the operator is unchanged by the transformation. (cid:3)
The remaining surgery operation deals with expanding the graph by inserting anew finite, connected metric graph Γ in some way to the original graph. If thereis a potential q associated with Γ , then we assume that it satisfies the followinghypothesis in agreement with what is assumed for q on Γ. Hypothesis 4.11.
On the finite, connected metric graph Γ , the potential q :Γ → R is measurable and bounded, and q ( x ) ≥ x ∈ Γ .As a rule, if no new potential is specified on the new edges, then it is reasonable totake the potential to be zero there. Nevertheless, the inequalities in Theorem 4.13hold for the potential chosen arbitrarily there under Hypothesis 4.11. Definition 4.12.
Let Hypothesis 3.1 be satisfied, and let e Γ be the graph formedfrom Γ by gluing m of the vertices of a finite, connected metric graph Γ to dis-tinct vertices of Γ. The new potential e q associated with e Γ is identical to q on theedges inherited from Γ and satisfies Hypothesis 4.11 on the edges from Γ . Thetransformation − ∆ K , Γ ,q
7→ − ∆ K , e Γ , e 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. • • •• • • • • •• Γ Γ • • •• • • • • e Γ Figure 5.
Attaching Γ to Γ by two vertices to form e Γ. The newedges in e Γ are shown in bold.
Theorem 4.13 (Attaching a graph) . Assume that Hypotheses 3.1 and 4.11 hold.Let e Γ be the graph formed by attaching Γ to Γ by m vertices. Then for the corre-sponding perturbed Krein Laplacians: (a) the positive eigenvalues satisfy e λ + j ≤ λ + j , j ∈ N ; (4.9)(b) the eigenvalues (counting ground states) satisfy e λ j + V − m ≤ λ j , j ∈ N ; (4.10) here V is the number of vertices of Γ .Proof. Every function in e H (Γ) can be extended by zero to a function in e H ( e Γ),and this does not change the Rayleigh quotient. Thus inequality (4.9) follows fromTheorem 3.11. Finally, (4.10) is obtained from (3.6), since the dimension of thekernel of the operator increases by V − m . (cid:3) A special case of the previous theorem consists of inserting a single edge betweentwo vertices of Γ, a process which does not change the dimension of the kernel ofthe perturbed Krein Laplacian.
Corollary 4.14 (Inserting an edge between existing vertices) . Let Hypothesis 3.1hold, and let e Γ be the graph formed by inserting an edge e between two (not nec-essarily distinct) vertices of Γ . Assume that the potential q on Γ = e satisfiesHypothesis 4.11. Then the eigenvalues (counting ground states) of the correspondingperturbed Krein Laplacians satisfy e λ j ≤ λ j , j ∈ N . We emphasise that this behaviour differs substantially from the one for stan-dard vertex conditions, where inserting an edge may either increase or decreaseeigenvalues; cf. [33]. 5.
Isoperimetric inequalities
We now turn to estimates for the positive eigenvalues of the perturbed KreinLaplacian. We start with a lower estimate for the first positive eigenvalue, whichwe may call the spectral gap ; cf. Remark 5.2 below.
Theorem 5.1.
Assume Hypothesis 3.1, and denote by ℓ (Γ) the total length of Γ .Furthermore, let Λ be the loop of length ℓ (Γ) . Then λ +1 ( − ∆ K , Γ ,q ) ≥ λ +1 ( − ∆ δ, Λ ,I ) (5.1) holds, where − ∆ Λ ,I is the Laplacian on Λ with a δ -interaction of strength I := R Γ q d x at one (arbitrary) point. In particular, λ +1 ( − ∆ K , Γ ) ≥ (cid:18) πℓ (Γ) (cid:19) . (5.2) Equality in (5.2) holds if and only if Γ is an interval, a loop, an equilateral 2-cycle,or an equilateral figure-8.Proof. Let e Γ be the flower graph formed from Γ by gluing all vertices. Then byTheorem 4.2 and Theorem 3.12, we have λ +1 ( − ∆ K , Γ ,q ) ≥ λ +1 ( − ∆ K , e Γ ,q ) ≥ λ +1 ( − ∆ st , e Γ ,q ) . HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 25
Moreover, as the only vertex of e Γ has even degree (equal to twice the number ofedges), we may cut through the vertex in such a way that we obtain an (Euler-ian) cycle Λ of length ℓ (Γ), and by surgery principles for the perturbed standardLaplacian − ∆ st , e Γ ,q , see e.g. [44, Theorem 4.1], we get λ +1 ( − ∆ st , e Γ ,q ) ≥ λ +1 ( − ∆ st , Λ ,q ) . In the case q = 0 identically, we have hereby shown (5.1), and (5.2) follows from adirect 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 inthe proof of [26, Theorem 1]: let ψ be an eigenfunction of − ∆ st , Γ ,q correspondingto its lowest eigenvalue. Then λ +1 ( − ∆ st , Λ ,q ) = R Γ | ψ ′ | d x + R Γ q | ψ | d x R Γ | ψ | d x ≥ R Γ | ψ ′ | d x + R Γ q | ψ ( x min ) | d x R Γ | ψ | d x , where x min is any point on Γ where | ψ | takes its minimum. Since the last quotientis the Rayleigh quotient of the Laplacian with a δ -vertex condition of strength I = R Γ q d x 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 beequalities. In particular, the standard Laplacian on the flower graph e Γ in the aboveargument already has to have 4 π /ℓ ( e Γ) as its first positive eigenvalue, which isonly possible if on the loop Λ resulting from splitting the central vertex of e Γ, thereexists an eigenfunction for the first positive eigenvalue which has the same valueat each point that was glued together previously (cf. [33, Theorem 1]). Since eacheigenfunction of − ∆ st , Λ corresponding to the first nonzero eigenvalue takes each ofits values exactly twice on the loop – at two points with distance ℓ (Γ) / e Γ can be recovered from Λ by gluing at most two points.Hence e Γ is either a loop itself or an equilateral figure-8. In other words, joiningall vertices in the original graph Γ leads to a loop or a figure-8, and this is onlypossible if Γ is of one of the following six types: an interval, a path graph with twoequal edges, a loop, an equilateral 2-cycle or an equilateral figure-8. Consideringthese graphs only, one finds by calculation that there exist eigenfunctions withcorresponding eigenvalue 4 π /ℓ (Γ) if and only if Γ is equilateral and has one ofthe four forms listed in the statement of the theorem. (cid:3) Remark 5.2.
The interval (0 , π /ℓ (Γ) ) has empty intersection not only with thespectrum of the Krein Laplacian on one individual graph Γ. In fact, Theorem 5.1asserts that, for fixed ℓ >
0, the interval (0 , π /ℓ ) is free of spectrum for the KreinLaplacians on the whole class of metric graphs with total length ℓ . Remark 5.3.
Alternatively, one may use (3.12) in combination with known lowerbounds on the eigenvalues of the standard Laplacian to obtain lower bounds forthe positive Krein Laplacian eigenvalues. However, using the optimal lower boundfrom [20], one gets λ +1 ( − ∆ K , Γ ) ≥ λ ( − ∆ st , Γ ) ≥ π ℓ (Γ) , which is weaker than the sharp bound (5.2). Remark 5.4.
The two crucial surgery operations used in the above proof are stan-dard: gluing all vertices of a graph into one was used in [29], and cutting throughvertices to obtain an Eulerian cycle goes back at least to [34]. Nevertheless, theabove proof is slightly unusual: for the standard Laplacian, gluing vertices increaseseigenvalues (the positive ones, as well as counting the ground state) whilst cuttingvertices decreases them, so that both surgery operations used above – gluing all vertices into one and cutting vertices to obtain an Eulerian cycle – cannot be usedwithin the same argument. However, in the present situation this works smoothlysince gluing is performed on the positive eigenvalues of the perturbed Krein Lapla-cian 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 eigen-values in the potential-free case:
Theorem 5.5.
Assume that Hypothesis 3.1 is satisfied with q = 0 identically, andthat Λ is a loop with the same length as for Γ . Then λ + j ( − ∆ K , Γ ) ≥ λ + j ( − ∆ st , Λ ) = λ j +1 ( − ∆ st , Λ ) holds for all j ∈ N . We conclude this section with a remark on how to apply the min-max principle toget 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 hereto the potential-free case, although natural generalisations for potentials exist (buttheir formulation may be less pleasant).
Remark 5.6.
Suppose that Γ contains an Eulerian cycle Σ (obtained by cuttingthrough vertices and removing edges not on the cycle), and let E Σ ⊆ E denote theset of edges belonging to Σ. Then the function f which on each e ∈ E Σ takes theform f e ( x ) = ± ℓ ( e ) n e sin (cid:18) n e πxℓ ( e ) (cid:19) , x ∈ [0 , ℓ ( e )] , 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 / f e on adjacent pairs of edges; the only place where there could be a discrepancyis when one returns to the start of the cycle, as the function may end on a half-number of periods, but this problem is averted by imposing the further restrictionthat P e ∈E Σ n e ∈ N . Now, f satisfies Dirichlet-Kirchhoff conditions not only onΣ, but also on Γ, after extending it by zero on E\E Σ , so such functions provideupper estimates for the positive eigenvalues of − ∆ K , Γ via the min-max principle,Theorem 3.11. The Rayleigh quotient for this f is R K [ f ] = π ℓ (Σ) X e ∈E Σ n e ℓ ( e ) , which is an explicit upper bound for the first positive eigenvalue; the maximumvalue of R K [ f ] among j linearly independent functions of this type gives an upperestimate for λ + j ( − ∆ K , Γ ).Of course, it is true in general, even with potentials, that for Γ containing anEulerian cycle Σ, one has λ + j ( − ∆ K , Γ ,q ) ≤ λ + j ( − ∆ K , Σ , e q ), where e q := q | Σ , due toTheorems 4.2 and 4.13.6. More general perturbed Krein Laplacians
Thus far, we have studied the Krein extension of the symmetric perturbed Lapla-cian with Dirichlet and Kirchhoff conditions at all vertices, but the abstract theoryof Krein extensions of symmetric operators allows one to extend this work to coversymmetric perturbed Laplacians with more general vertex conditions. In this sec-tion we illustrate this by considering perturbed Laplacians with “Krein vertex con-ditions” 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
HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 27 the previous sections carry over to this setting. The proofs are analogous in thepresent case and are mostly left to the reader.Let Hypothesis 3.1 be satisfied. For
B ⊂ V , define the operator S B in L (Γ) by( S B f ) e = − f ′′ e + q e f e on each edge e ∈ E , dom S B = n f ∈ e H (Γ) ∩ H (Γ) : ∂ ν f ( v ) = 0 for each v ∈ V ,f ( v ) = 0 for each v ∈ B o . (6.1) Remark 6.1.
A more general setting may be treated with the same methods, butwe do not go into these details here: it is possible to replace the standard vertexconditions at the vertices in
V \ B by any self-adjoint, local vertex conditions. Forthe 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 ofboundary for Γ. One choice, which may be natural in some cases, is to let B consistof all vertices of degree one. We are not restricted to this situation, but we maykeep it in mind as a typical example.The operator S B in (6.1) is symmetric, closed, and densely defined. It has defectnumbers n − = n + = |B| , and is thus only self-adjoint if B = ∅ . Furthermore, S B isclearly nonnegative, and its Friedrichs extension S B , F is the perturbed Krein Lapla-cian subject to Dirichlet boundary conditions on B and standard vertex conditionson V \ B . In particular, ( S B f, f ) ≥ µ k f k , f ∈ dom S B (6.2)holds, where µ > S B , F . The adjoint of S B equals( S ∗B f ) e = − f ′′ e + q e f e on each edge e ∈ E , dom S ∗ = n f ∈ e H (Γ) ∩ H (Γ) : ∂ ν f ( v ) = 0 for all v ∈ V \ B o . Due to (6.2), for nonempty
B ⊂ V , we may consider the operator − ∆ K , Γ ,q, B := S B , K , the Krein–von Neumann extension of S B . If q = 0 identically, we write − ∆ K , Γ , B .To derive some properties of the operator − ∆ K , Γ ,q, B , constructing an appropriateboundary triple is useful. Proposition 6.3.
Assume that Hypothesis 3.1 is satisfied, and let
B ⊂ V benonempty. Let S B be defined in (6.1) . For f ∈ dom S ∗B , define Γ f = f ( v ) ... f ( v b ) and Γ f = − ∂ ν f ( v ) ... − ∂ ν f ( v b ) , where B = { v , . . . , v b } (and b = |B| ). Then { C b , Γ , Γ } is a boundary triple for S ∗B ; in particular, S B has defect numbers n − = n + = b. The corresponding extensions A and B of S defined in (2.14) are given by A = S B , F and B = − ∆ st , Γ ,q ; in particular, ∈ ρ ( A ) . The value of the corresponding Weyl function at λ = 0 is M B (0) = − Λ q, B , where Λ q, B is the Dirichlet-to-Neumann matrix for B defined viathe relation ∂ ν f ∗ ( v ) ... ∂ ν f ∗ ( v b ) = Λ q, B f ∗ ( v ) ... f ∗ ( v b ) , where f ∗ ∈ ker S ∗B is arbitrary. Remark 6.4.
The Weyl function λ M B ( λ ) may be computed from the Weylfunction λ M ( λ ) of the boundary triple in Proposition 3.3. Indeed, if we write V = { v , . . . , v b , v b +1 , . . . , v V } , where the vertices are ordered such that the first b of them form B , and write M ( λ ) = b D ( λ ) − B ( λ ) ⊤ − B ( λ ) b L ( λ ) ! , where the block decomposition is taken according to the decomposition of the ver-tices into B and V \ B , then we have M B ( λ ) = b D ( λ ) − B ( λ ) ⊤ b L ( λ ) − B ( λ ) . The proof is straightforward; for a special case it may be found in [22, Proposi-tion 3.1]; see also [30, Lemma 3.1]. In particular, in the potential-free case, − M B (0)is the Schur complement of the weighted discrete Laplacian L in (3.5) with respectto decomposition of the vertices into B and V \ B .From Proposition 6.3, the following properties of − ∆ K , Γ ,q, B are immediate. Proposition 6.5.
Assume that Hypothesis 3.1 holds and that
B ⊂ V is nonempty.Then − ∆ K , Γ ,q, B acts as (cid:0) − ∆ K , Γ ,q, B f (cid:1) e = − f ′′ e + q e f e on each edge e ∈ E , and its domain consists of all f ∈ e H (Γ) ∩ H (Γ) such that ∂ ν f ( v ) ... ∂ ν f ( v d ) = (cid:0) b D − B ⊤ b L − B (cid:1) f ( v ) ... f ( v d ) , where we have written Λ q, B = b D − B ⊤ − B b L ! , in block matrix form with respect to the decomposition of V into B and V \ B .Moreover, dim ker (cid:0) − ∆ K , Γ ,q, B (cid:1) = dim ker S ∗B = b. Next, as an application of the abstract Theorem 2.4, we obtain the followingvariational characterisation for the eigenvalues of − ∆ K , Γ ,q, B . Theorem 6.6.
If Hypothesis 3.1 is satisfied and
B ⊂ V is nonempty, then thespectrum of − ∆ K , Γ ,q, B is purely discrete, and the positive eigenvalues λ +1 (cid:0) − ∆ K , Γ ,q, B (cid:1) ≤ λ +2 (cid:0) − ∆ K , Γ ,q, B (cid:1) ≤ . . . of − ∆ K , Γ ,q, B , counted with multiplicities, satisfy λ + j (cid:0) − ∆ K , Γ ,q, B (cid:1) = min F ⊂ dom S B dim F = j max f ∈ Ff =0 R Γ |− f ′′ + qf | d x R Γ | f ′ | d x + R Γ q | f | d x HE KREIN–VON NEUMANN EXTENSION FOR METRIC GRAPHS 29 for all j ∈ N . 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 par-ticular, one gets the following. Theorem 6.7.
Assume that Hypothesis 3.1 is satisfied and that
B ⊂ V is nonempty.Then dim ran h(cid:0) − ∆ K , Γ ,q, B − λ (cid:1) − − (cid:0) − ∆ st , Γ ,q − λ (cid:1) − i = dim ran Λ q, B = ( b − if q = 0 identically ,b, else , where b = |B| . In particular, in the potential-free case, if b = |B| = 1, then − ∆ K , Γ , B equals thestandard Laplacian. As a consequence of either Theorem 6.7 or Theorem 6.6, weget, analogously to (3.11), λ j +1 (cid:0) − ∆ st , Γ (cid:1) ≤ λ + j (cid:0) − ∆ K , Γ , B (cid:1) = λ j + b (cid:0) − ∆ K , Γ , B (cid:1) , j ∈ N , in the case without potential.The surgery principles of Section 4 remain valid for the (positive) eigenvalues ofthe operator − ∆ K , Γ ,q, B , provided that all vertices involved in the surgery operationsbelong to B ; we leave it to the reader to formulate and prove the corresponding re-sults. Instead we formulate a related result which deals with the transition betweenstandard and “Krein vertex conditions”. Theorem 6.8.
Let Hypothesis 3.1 be satisfied. Moreover, let e B ⊂ B ⊂ V be sets ofsize b = |B| and e b = | e B| , respectively, and let k := b − e b . Then for λ + j := λ + j (cid:0) − ∆ K , Γ ,q, B (cid:1) , e λ + j := λ + j (cid:0) − ∆ K , Γ ,q, e B (cid:1) ,λ j := λ j (cid:0) − ∆ K , Γ ,q, B (cid:1) , e λ j := λ j (cid:0) − ∆ K , Γ ,q, e B (cid:1) , the following statements hold: (i) the positive eigenvalues satisfy e λ + j ≤ λ + j ≤ e λ + j + k ≤ λ + j + k , j ∈ N ; (6.3)(ii) the eigenvalues (counting ground states) satisfy λ j ≤ e λ j ≤ λ j + k ≤ e λ j + k , j ∈ N . (6.4) Proof.
If we denote by S and e S the symmetric operators defined in (6.1) for thevertex subsets B and e B respectively, then e B ⊂ B implies the operator inclusion S ⊂ e S . Moreover, it is easy to see that dom S has co-dimension k = b − e b indom e S . Therefore (6.3) follows directly from Theorem 2.8. Using the fact thatthe perturbed Krein Laplacians for B and e B have respectively b and e b linearlyindependent functions in their kernels, (6.4) follows from (6.3). (cid:3) As a simple consequence, for any nonempty
B ⊂ V , we have λ + j (cid:0) − ∆ st , Γ ,q (cid:1) ≤ λ + j (cid:0) − ∆ K , Γ ,q, B (cid:1) ≤ λ + j (cid:0) − ∆ K , Γ ,q (cid:1) as well as λ j (cid:0) − ∆ K , Γ ,q (cid:1) ≤ λ j (cid:0) − ∆ K , Γ ,q, B (cid:1) ≤ λ j (cid:0) − ∆ st , Γ ,q (cid:1) , for all j ∈ N . Acknowledgement.
The authors are grateful to Fritz Gesztesy for his commentson the literature. J.R. acknowledges financial support by grant no. 2018-04560 ofthe Swedish Research Council (VR).
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