A note on the Gaffney Laplacian on infinite metric graphs

We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.


Introduction
The standard way to associate an operator with the Laplacian on an unbounded domain in R n or on a non-compact Riemannian manifold is to consider it either on smooth compactly supported functions (the so-called pre-minimal operator) or on the largest possible natural domain when Laplacian is understood in a distributional sense in L 2 (the maximal operator). Then the pre-minimal Laplacian is essentially self-adjoint if and only if its L 2 closure, the minimal operator, coincides with the maximal operator. For geodesically complete manifolds the essential selfadjointness was proved by W. Roelcke [28] (see also [6,32]). It is impossible to give even a brief account on the subject and we only refer to [2,8,30], some recent work in the case of non-complete manifolds [24,25] and also for weighted graph Laplacians [17,20]. If we further restrict the maximal operator to functions having finite energy, the self-adjointness of this operator (called the Gaffney Laplacian [11,16]) is equivalent to the uniqueness of a Markovian extension. Clearly, essential selfadjointness implies the uniqueness of Markovian extensions, but the converse is not necessarily true. We refer to [8,Chapter I] for an excellent account on importance and applications of both the self-adjoint and Markovian uniqueness.
The main object of our paper is a quantum graph, i.e., a Laplacian on a metric graph. From the perspective of Dirichlet forms, quantum graphs play an important role as an intermediate setting between Laplacians on Riemannian manifolds and discrete Laplacians on weighted graphs. The most studied quantum graph operator is the Kirchhoff Laplacian, which provides the analog of the Laplace-Beltrami operator in the setting of metric graphs. Whereas on finite metric graphs the Kirchhoff Laplacian is always self-adjoint, the question is more subtle for graphs with infinitely many edges. Geodesic completeness (w.r.t. the natural path metric) guarantees self-adjointness of the (minimal) Kirchhoff Laplacian, however, this result is far from being optimal [9, §4]. The present paper is a complement to the recent work [22], where a relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph was established. More precisely, the notion of finite volume for ends of a metric graph introduced in [22] turns out to be the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. Our main goal is to elaborate on the relationship between finite volume graph ends and the Gaffney Laplacian, which is defined as the restriction of the maximal Kirchhoff Laplacian to functions having finite energy (i.e., H 1 functions). First of all, one of the main results of [22] provides a transparent geometric characterization of the self-adjointness of the Gaffney Laplacian: the underlying metric graph has no finite volume ends (see Lemma 3.6 below). Our first main result shows that the deficiency indices of the minimal Gaffney Laplacian (i.e., the degree of the unitary group parameterizing its self-adjoint extensions) are in fact equal to the number of finite volume graph ends (Theorem 3.11). The Gaffney Laplacian has several advantages comparing to the maximal Kirchhoff Laplacian, although one of the main disadvantages is the fact that it is not necessarily closed. Our second main result, Theorem 3.12 provides necessary and sufficient conditions for the Gaffney Laplacian to be closed (see also Prop. 4.1). These conditions are stated in terms of finite volume graph ends and in certain cases of interest (graphs of finite total volume or Cayley graphs of countable finitely generated groups) they give rise to a transparent geometrical criterion: the Gaffney Laplacian is closed if and only if the underlying metric graph has finitely many finite volume ends, which is further equivalent to the fact that the deficiency indices of the minimal Gaffney Laplacian are finite. If the Gaffney Laplacian is not closed, then the most important question is how to describe its L 2 closure. It does not seem realistic to us to obtain a complete answer to this question and we demonstrate by examples that under certain symmetry conditions the closure of the Gaffney Laplacian may coincide with the maximal Kirchhoff Laplacian.
Let us now briefly describe the structure of the article. Section 2 is of preliminary character where we collect basic notions and facts about graphs and metric graphs (Section 2.1); graph ends (Section 2.2); Sobolev spaces on metric graphs (Section 2.3); Kirchhoff, Dirichlet and Neumann Laplacians on metric graphs (Section 2.4). The main results of the present paper are collected in Section 3. We introduce the minimal Gaffney Laplacian and the Gaffney Laplacian, study their properties and also investigate their relationship with the Kirchhoff, Dirichlet and Neumann Laplacians. In the final section we discuss several explicit examples.
Notation. Z, R, C have their usual meaning; Z ≥a := Z ∩ [a, ∞). z * denotes the complex conjugate of z ∈ C. For a given set S, #S denotes its cardinality if S is finite; otherwise we set #S = ∞. If it is not explicitly stated otherwise, we shall denote by (x n ) a sequence (x n ) ∞ n=0 .
2. Quantum graphs and graph ends 2.1. Combinatorial and metric graphs. In what follows, G d = (V, E) will be an unoriented graph with countably infinite sets of vertices V and edges E. For two vertices u, v ∈ V we shall write u ∼ v if there is an edge e u,v ∈ E connecting u with v. For every v ∈ V, we denote the set of edges incident to the vertex v by E v and The following assumption is imposed throughout the paper.
Hypothesis 2.1. G d is simple (no loops or multiple edges) 1 , locally finite (deg(v) < ∞ for every v ∈ V) and connected (for any u, v ∈ V there is a path connecting u and v).
Assigning to each edge e ∈ E a finite length |e| ∈ (0, ∞) turns G d into a metric graph G := (V, E, | · |) = (G d , | · |). The latter equips G with a (natural) topology and metric. More specifically (see, e.g., [18, Chapter 1.1]), a metric graph G is a Hausdorff topological space with countable base such that each point x ∈ G has a neighbourhood E x (r) homeomorphic to a star-shaped set E(deg(x), r x ) of degree deg(x) ≥ 1, Identifying every edge e ∈ E with a copy of an interval of length |e| and also identifying the ends of the edges that correspond to the same vertex v, G can be equipped with the natural path metric ̺ -the distance between two points x, y ∈ G is defined as the length of the "shortest" path connecting x and y.

Graph ends.
A sequence of distinct vertices (v n ) such that v n ∼ v n+1 for all n ∈ Z ≥0 is called a ray. Two rays R 1 , R 2 are called equivalent if there is a third ray containing infinitely many vertices of both R 1 and R 2 . An equivalence class of rays is called a graph end of G d .
Considering a metric graph G as a topological space, one can introduce topological ends. Consider sequences U = (U n ) of non-empty open connected subsets of G with compact boundaries and such that U n+1 ⊆ U n for all n ≥ 0 and n≥0 U n = ∅. Two such sequences U and U ′ are called equivalent if for all n ≥ 0 there exist j and k such that U n ⊇ U ′ j and U ′ n ⊇ U k . An equivalence class γ of sequences is called a topological end of G and C(G) denotes the set of topological ends of G. There is a natural bijection between topological ends of a locally finite metric graph G and graph ends of the underlying combinatorial graph G d : for every topological end γ ∈ C(G) of G there exists a unique graph end ω γ of G d such that for every sequence U representing γ, each U n contains a ray from ω γ (see [33, §21], [7, §8.6 and also p.277-278] for further details).
One of the main features of graph ends is that they provide a rather refined way of compactifying graphs, called the end (or Freudenthal) compactification of G (see [7, §8.6], [33] and also [22, §2.2]). Definition 2.1. An end ω of a graph G d is called free if there is a finite set X of vertices such that X separates ω from all other ends of the graph.
Remark 2.2. Notice that an end γ ∈ C(G) is free exactly when there exists a connected subgraph G with compact boundary ∂ G 2 such that U n ⊆ G eventually for any sequence U = (U n ) representing γ and U ′ n ∩ G = ∅ eventually for all sequences U ′ = (U ′ n ) representing an end γ ′ = γ. We also need the following notion introduced in [22].
The set of all finite volume ends is denoted by C 0 (G).

2.3.
Function spaces on metric graphs. Identifying every edge e ∈ E with the copy of I e = [0, |e|] (and hence assigning an orientation on G), we can introduce Sobolev spaces on edges and on G. First of all, the Hilbert space L 2 (G) of functions f : G → C is defined by The subspace of compactly supported L 2 (G) functions will be denoted by only on finitely many edges e ∈ E . For edgewise locally absolutely continuous functions on G let us denote by ∇ the edgewise first derivative, ∇f := f ′ .
(2.2) Then for every edge e ∈ E, where AC(e) is the space of absolutely continuous functions on e. Next H n (G \ V), n ∈ {1, 2} is defined as the space of functions f : G → C such that It becomes a Hilbert space when equipped with the norm f 2 which is also a Hilbert space when equipped with the above norm. We also define [22,Lemma 3.2]) and, moreover, every function f ∈ H 1 (G) admits a unique continuous extension to the end compactification G of G ( [22,Prop. 3.5]): for every f ∈ H 1 (G) and a (topological) end γ ∈ C(G), we define where R = (v n ) ∈ ω γ is any ray belonging to the corresponding graph end ω γ . It turns out that finite volume graph ends serve as a proper boundary for the Sobolev space H 1 (G). Namely, considering H 1 (G) as a subalgebra of C b (G), it was proved in [22, §3] that its closure is isomorphic to C 0 (G ∪ C 0 (G)). In particular, ends having infinite volume lead to trivial values, that is, f (γ) = 0 3 As usual, vol(A) denotes the Lebesgue measure of a measurable set A ⊆ G. for every f ∈ H 1 (G) if and only if γ / ∈ C 0 (G). Moreover, by [22,Theorem 3.10], and hence H 1 (G) = H 1 0 (G) exactly when G has no finite volume ends, C 0 (G) = ∅.

2.4.
Kirchhoff, Dirichlet and Neumann Laplacians. Let G be a metric graph satisfying Hypothesis 2.1. In the Hilbert space L 2 (G), we can introduce the Laplacian ∆ defined on the maximal domain H 2 (G \ V) and acting on each edge as the negative second derivative, ∆ = −∇ 2 . If v is a vertex of the edge e ∈ E, then for every f ∈ H 2 (e) the quantities are well defined. The Kirchhoff (also called standard or Kirchhoff-Neumann) boundary conditions at every vertex v ∈ V are then given by Imposing these boundary conditions on the maximal domain yields the maximal Kirchhoff Laplacian Restricting further to compactly supported functions we end up with the preminimal operator We call its closure H 0 := H 0 0 in L 2 (G) the minimal Kirchhoff Laplacian. Integrating by parts one obtains and hence both H 0 0 and H 0 are non-negative symmetric operators. It is known that The equality H 0 = H holds if and only if H 0 is self-adjoint (or, equivalently, H 0 0 is essentially self-adjoint). To the best of our knowledge, the strongest sufficient condition which guaranties self-adjointness is provided by the next result.
). Let ̺ m be the star path metric on V, If (V, ̺ m ) is complete as a metric space, then H 0 0 is essentially self-adjoint.
The degree of non-self-adjointness of H 0 is determined by its deficiency indices is closed (since H 1 (G) is a Hilbert space). The self-adjoint operator H N associated with this form is usually called the Neumann extension of H 0 (not to confuse it with Krein-von Neumann extension!) or Neumann Laplacian.
Remark 2.5. Following the analogy with the Dirichlet Laplacian, it might be tempting to conjecture that the domain of the Neumann Laplacian is given by dom(H) ∩ H 1 (G). However, the operator defined on this domain has a different name -the Gaffney Laplacian -and it is not even symmetric in general. The main focus of the following two sections will be on the study of this operator.

The Gaffney Laplacian
Let us fix an orientation on G. In the Hilbert space L 2 (G), we can associate (at least) two gradient operators with ∇ defined by (2.2). Namely, set The importance of ∇ D and ∇ N stems from the following fact.
Proof. Since Now we are in position to introduce the main object. In the Hilbert space L 2 (G), define the following operators They act edgewise as the negative second derivative and their domains are The operator H G is called the Gaffney Laplacian. We shall refer to H G,min as the minimal Gaffney Laplacian. Remark 3.3. Notice that the above definition is not precisely the original definition of M. P. Gaffney [11] (roughly speaking H 1 was replaced in [11,12] by C 1 ∩ H 1 , see also [24,25]). The obvious drawback is that the corresponding maximal Laplacian is always non-closed. Let us also stress that we are unaware of H G,min in the manifold context and this natural, in our opinion, object seems to be new.
Remark 3.4. One can introduce 0-forms and 1-forms on G and, upon assigning an orientation, both can be further identified with functions (see, e.g., [27]). 5 From this perspective the operator ∆ = ∇ N ∇ * D is a metric graph analogue of the Hodge Laplacian on 1-forms (see [1, §5.1], [13], [27]). Indeed (see, e.g., [15]), the Hodge Laplacian on smooth k-forms is given by where d k is the exterior derivative (mapping k-forms to (k + 1)-forms) and the codifferential δ k+1 is its formal adjoint (mapping (k + 1)-forms to k-forms). Working in the L 2 -framework and replacing smooth by H 1 for metric graphs, one can identify d 1 = ∇ N and δ 0 = ∇ * D . In particular, the Gaffney Laplacian (3.4) can be viewed as the Hodge Laplacian on 0-forms. Let us also stress that due to the supersymmetry, the properties of H G and ∆ are closely connected.
Proof. It is straightforward to verify both claims, however, we would like to show that which then makes the inclusions in (3.5) obvious. Here we employ the following notation If f belongs to the RHS in (3.6), then an integration by parts gives Clearly, ℓ f extends to a bounded linear functional on L 2 , which implies that f ∈ dom(∇ * D ) and hence this proves the inclusion "⊇". Suppose now that f ∈ dom(∇ * D ). Fixing an edge e ∈ E and taking a test function g ∈ H 1 0 (G) such that g equals zero everywhere except e, we immediately conclude that f belongs to H 1 on e. Next pick a vertex v ∈ V. Choose g ∈ H 1 0 (G) such that g ≡ 0 on E \ E v . Moreover, for every e ∈ E v we assume that g(x e ) = 1 if x e ∈ e and |x e − v| < |e|/4 and g(x e ) = 0 if |x e − v| > |e|/2. Thus we get This also implies that we can perform the integration by parts in (3.7) for every . Since ℓ f extends to a bounded linear functional on L 2 (G), we conclude that ∇f ∈ L 2 (G), which completes the proof.
Clearly, all four operators in (3.5) coincide exactly when H 0 is self-adjoint (and hence all four operators are self-adjoint). Moreover, by the very definition we have In particular, H G,min is symmetric, however, H G may not be symmetric (and hence self-adjoint). The next result provides several self-adjointness criteria for H G including a transparent geometric characterization.
Lemma 3.6. The following statements are equivalent:  H 1 (G). In particular, every Markovian extension is a finite energy extension (for further details we refer to [22, §5]).
The importance of the Gaffney Laplacian in the study of Markovian and, more generally, finite energy extensions of H 0 stems from the following fact.  . Let us also stress that the proper analog of the equality ∇ N = ∇ D in the manifold context should read as "the gradient is a formal skew-adjoint of the divergence operator" (see Remark 3.4) and this property was called negligible boundary in [11,12]. Moreover, the central result of [11] is that negligible boundary is sufficient for the (essential) self-adjointness of the Gaffney Laplacian (notice that the quadratic forms approach was not available at that time).
As in the case of the maximal and minimal Kirchhoff Laplacians, there is a close connection between the Gaffney Laplacians.
which implies (3.10). Since the adjoint is always closed, the first claim follows. Indeed, one simply needs to compare the definition of H G,min with (3.3) and take into account that ∇ * N ⊆ ∇ * D . Since both H D and H N are closed, one concludes that so is H G,min .
Our first main result is the following connection between deficiency indices of H G,min and graph ends. Proof. If #C 0 (G) < ∞, then using the results of [22, §6] one can easily see that dim(dom(H N )/ dom(H G,min )) = #C 0 (G) (indeed, combine Prop. 6.6(i) and Corollary 6.7 with Theorem 3.11 from [22]). This immediately implies (3.12) since H N is a self-adjoint extension of H G,min (cf. [29,Theorem 13.10]). It remains to consider the case #C 0 (G) = ∞. By (3.9) and [22,Lemma 4.7], for all negative real λ. It remains to notice that H G ⊆ H * G,min and hence the claim follows from the positivity of H G,min .
In contrast to the minimal Gaffney Laplacian, H G is not automatically closed, that is, H * G,min = H G , although it is not necessarily true that H * G,min = H G . (3.14) Our second main result provides necessary and sufficient conditions for the Gaffney Laplacian to be closed. (ii) Since the mapping ∇ : H 1 (G) → L 2 (G) is bounded, (3.14) holds if and only if there exists a positive constant C > 0 such that Suppose G has a non-free end of finite volume γ and fix a sequence U = (U n ) of open sets representing γ. We can choose a sequence (G n ) of connected subgraphs of G such that #∂G n < ∞, almost all of the open sets U k are contained in G n and G n ⊂ U n for all n. The latter two properties imply that #C 0 (G n ) = ∞ for every n ≥ 0 and n≥0 G n = ∅. In particular, vol(G n ) → 0 as n → ∞.
Fix λ < 0 and denote by H n the Gaffney Laplacian on the subgraph G n , n ≥ 0. Taking into account (3.13), there exists a real-valued function f n ∈ ker(H n − λ) with f n (v) = 0 for all v ∈ ∂G n and such that f n ∞ = 1 Moreover, extending f n by zero on G \ G n gives a function (also denoted by f n ) belonging to the domain of the Gaffney Laplacian H G on G.
Assuming that H G is closed, (3.16) would imply that for all n ≥ 0. Next, for each n there is x n ∈ G n with |f n (x n )| ≥ 1/2. Choosing y n ∈ ∂G n and a path P n in G n connecting x n and y n , we get for all n ≥ 0. However, the right-hand side tends to zero when n → ∞. This contradiction completes the proof.
Let us now present two particular cases of interest when Theorem 3.12 provides a necessary and sufficient condition for H G to be closed.
Corollary 3.13. Suppose G has finite total volume. The following are equivalent: (i) The Gaffney Laplacian H G is closed, (ii) (3.14) holds true, (iii) G has finitely many ends, #C(G) < ∞.
Proof. We only need to notice that C(G) = C 0 (G) in the case when vol(G) < ∞. By Halin's theorem [19], a locally finite graph G has at least one end which is not free if C(G) = ∞. Thus, it remains to apply Theorem 3.12.
Theorem 3.12 also gives rise to a criterion in the case of Cayley graphs.
Corollary 3.14. Suppose G d is a Cayley graph of a countable finitely generated group G. Then H G is not closed if and only if #C(G) = ∞ and G has at least one finite volume end.
Proof. If there are infinitely many ends, then the end space is known to be homeomorphic to the Cantor set (see, e.g., [14, Addendum 13.5.8]), and hence there are no free graph ends. Theorem 3.12 completes the proof.
Remark 3.15. By the Freudenthal-Hopf theorem, a Cayley graph of a countable finitely generated group has 1, 2 or infinitely many ends. Moreover, the number of ends is independent of the choice of the finite generating set. By Hopf's theorem, G d has exactly two ends if and only if G is virtually infinite cyclic. The classification of finitely generated groups with infinitely many ends is due to J. R. Stallings (see, e.g., [14,Chapter 13]). In particular, if G is amenable, then it has finitely many ends (actually, either 1 or 2) and hence the Gaffney Laplacian is always closed for Cayley graphs of amenable groups.
Remark 3.16. The above considerations shed more light on the results obtained in [22]. (i) First of all, combining (3.5) with (3.9) and Theorem 3.12(i) we obtain one of the main results in [22], Theorem 4.1, on the deficiency indices of the Kirchhoff Laplacian:

Examples
The case not covered by Theorem 3.12 is when G has infinitely many finite volume free ends, however, all non-free ends have infinite volume. Moreover, there is one additional problem: it is not clear what is the closure of H G if it is not closed. We begin with the following result. Proof. It is easy to see that properties (a) and (b) are exactly the ones used in the proof of Theorem 3.12(ii) (in the case of a non-free finite volume end #∂G n < ∞ and #C(G n ) = #C 0 (G n ) = ∞ for all n ≥ 0) and hence the proof of Proposition 4.1 is literally the same and we leave it to the reader. Proposition 4.1 enables us to construct graphs without finite volume non-free ends, however with the corresponding Gaffney Laplacian H G not being closed. Example 4.3. Take a path graph P 0 = (Z ≥0 , | · |) equipped with some positive lengths and attach to each vertex v n ∈ Z ≥0 an infinite rooted graph G n . If each G n has at least two ends (e.g., each G n consists of two rooted antitrees joined at the root vertices) and lim inf n vol(G n ) = 0, then H G is not closed (see Figure 1).   Figure 1. Assuming that each G n has finitely many graph ends and finite total volume, the Gaffney Laplacian H n on G n is closed, which is further equivalent to the validity of the Sobolev-type inequality (3.16) with some constant C n > 0, n ≥ 0. Also one has similar inequalities on every edge of a path graph, however, the corresponding constants do depend on the edges lengths (see, e.g.,  ds, x ∈ [t n , L).
With respect to the decomposition (4.1), every g n as well as ½ [0,L) defines a function in ker(H). In particular, ½ [0,L) gives rise to ½ T , which is clearly in H 1 (T ). Since Remark 4.7. Radially symmetric trees are a particular example of the so-called family preserving metric graphs (see [4] and also [3]). Employing the results from [4] (and also assuming no horizontal edges), it is in fact possible to show that for family preserving metric graphs H 0 = H G,min and hence either H G is closed (which holds exactly when the corresponding metric graph either has infinite volume, and hence H 0 is self-adjoint [22,Remark 4.12], or it has finite volume and finitely many ends) or its closure coincides with the maximal Kirchhoff Laplacian.