Coupling of symmetric operators and the third Green identity

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains on smooth, boundaryless Riemannian manifolds.


Introduction
The work in this paper is motivated in part by the following innocuous question: How to rule out that a Dirichlet eigenfunction of the Laplacian, or, more generally, a uniformly elliptic second order partial differential operator on a bounded domain Ω ⊂ R n , n ∈ N, is also simultaneously a Neumann eigenfunction?
Thus, vanishing of the normal derivative (i.e., the Neumann boundary condition) ∂u/∂ν = 0 on ∂Ω yields λ = 0 which contradicts the well-known fact that the Dirichlet Laplacian is strictly positive on bounded (in fact, finite Euclidean volume) domains Ω. A second approach is based on the fact that if Ω ⊂ R n is a bounded Lipschitz domain then (see [32,Corollary 3.21,p. 161] for more general results of this flavor) (1. 4) In turn, this may be used to show that, whenever Ω ⊂ R n is a bounded Lipschitz domain, if − ∆u = λu, u ∂Ω = (∂ ν u) ∂Ω = 0, u ∈ H 1 (Ω), (1.5) then u necessarily vanishes. Specifically, (1.5) first implies (see [4]) that u ∈ H 2 (Ω) which, together with (1.4), proves that the extension u of u to R n by zero outside Ω lies in H 2 (R n ) and satisfies −∆ u = λ u in R n . As such, if u were not identically zero in Ω, the Laplacian in R n would have a compactly supported eigenfunction, clearly a contradiction. This rules out the eventuality of a Dirichlet eigenfunction simultaneously being a Neumann eigenfunction. An argument of this nature may be adapted to more general uniformly elliptic second order partial differential operators via unique continuation principles; see, for instance, [2], [3], [20], [23], [25], [26], and [38].
An approach, intimately related to the one just discussed, adding a functional analytic flavor, would employ the fact that if Ω ⊂ R n is a bounded Lipschitz domain then the Dirichlet and Neumann Laplacians in L 2 (Ω), denoted by −∆ D,Ω and −∆ N,Ω , respectively, are relatively prime and hence satisfy (see, for instance, [4,Sect. 3]). Invoking the fact that the minimal operator −∆ min,Ω is simple (i.e., it has no invariant subspace on which it is self-adjoint), and simple operators have no eigenvalues, −∆ min,Ω cannot have any eigenvalues, thus, no nonzero solution u satisfying (1.5) exists. For recent results of this type see, for instance, [9,Proposition 2.5]. Upon modifications employing appropriate Dirichlet and Neumann traces this approach remains applicable to the more general case of uniformly elliptic second order partial differential operators on Lipschitz domains Ω (see, e.g., [4], [9]). Perhaps, a most illuminating proof of the impossibility of a Dirichlet eigenfunction to be simultaneously a Neumann eigenfunction can be based on the third Green identity, which naturally leads to one of the principal topics of this paper. Assuming again Ω to be a bounded Lipschitz domain in R n , we note the following well-known special case of the third Green identity (see, e.g., [12], [24, p. 97], [29,Theorem 6.10]), u ∈ H 2 (Ω), z ∈ C, x ∈ Ω, (1.10) in terms of the resolvent operator G z for −∆ defined on H 2 (R n ), and the single and double layer potentials S z and D z , z ∈ C, defined by Here E (0) n (z; x) represents the standard fundamental solution of the Helmholtz differential expression −∆ − z in R n , n ∈ N, n ≥ 2, that is, (n−2)/2 z 1/2 |x| , n ≥ 2, z ∈ C\{0}, −1 2π ln(|x|), n = 2, z = 0, 1 (n−2)ωn−1 |x| 2−n , n ≥ 3, z = 0, Im z 1/2 ≥ 0, x ∈ R n \{0}, (1.14) where H Thus, if u is assumed to satisfy (1.5) then, as already noted, u ∈ H 2 (Ω) so the third Green identity (1.10) instantly yields u = 0 in Ω, hence the nonexistence of nontrivial solutions u satisfying (1.5). Again, this approach extends to the more general case of uniformly elliptic second order partial differential operators L by appropriately replacing in (1.10)-(1.14) the fundamental solution E The approach based on Green's third identity just outlined has several features. First, while the original issue pertaining to properties of eigenfunctions of the Dirichlet Laplacian −∆ D,Ω in L 2 (Ω) may be formulated entirely in terms on Ω and ∂Ω, the third Green identity (1.10) naturally involves the fundamental solution E (0) n (z; x − y) for the Laplacian in R n . The latter is obviously logically independent of Ω and ∂Ω.
Second, denoting Ω + := Ω, Ω − := R n \Ω, (1.15) one is naturally led to a comparison of the Dirichlet Laplacian (1.16) and the Laplacian −∆ in L 2 (R n ) (with domain H 2 (R n )). While the Dirichlet Laplacian (−∆ D,Ω+ ) ⊕ (−∆ D,Ω− ) corresponds to a complete decoupling of R n into Ω + ∪ Ω − (ignoring the compact boundary C := ∂Ω ± of n-dimensional Lebesgue measure zero), in stark contrast to this decoupling, the Laplacian −∆ on H 2 (R n ) couples Ω + and Ω − via the imposition of continuity conditions across C of the form Here we identified u ∈ L 2 (R n ) with the pair (u + , u − ) ∈ L 2 (Ω + ) ⊕ L 2 (Ω − ) via u ± = u Ω± . The relative sign change in the normal derivatives in the second part of (1.17) is of course being a side-effect of the opposite orientations of the outward unit normals to Ω ± at points in C = ∂Ω ± . It is this coupling of Ω + and Ω − through their joint boundary C via the Laplacian −∆ on R n via the continuity requirements (1.17) which constitutes the second major topic in this paper.
In fact, from this point of view, the exterior domain Ω − plays a similar role as the original domain Ω = Ω + (apart from being unbounded). Moreover, introducing the jumps of u and ∂ ν u across C = ∂Ω ± as the third Green identity (1.10) can be shown to acquire the following form (which is symmetric with respect to Ω ± ; cf., e.g., [12], [29, Theorem 6.10]): At this point we are prepared to describe the major objectives of this paper: Decompose a given complex, separable Hilbert space H into an orthogonal sum of closed subspaces H ± as H = H + ⊕ H − , consider densely defined, closed symmetric operators S ± in H ± and their direct sum S = S + ⊕S − in H, introduce restrictions T ± of S * ± such that T ± = S * ± and appropriate restrictions A 0,± of T ± , for instance, A 0,± self-adjoint in H ± , defined in terms of certain abstract boundary conditions, and then find a self-adjoint operator A in H which closely resembles A 0 = A 0,+ ⊕ A 0,− , but without any remnants of the boundary conditions in A 0,+ ⊕ A 0,− and without any reference to the decomposition of H into H + ⊕H − (i.e., A naturally couples H ± in terms of certain continuity requirements through an abstract "boundary"). Finally, derive an abstract third Green identity invoking the resolvent (or fundamental solution operator G) of A, the operator T = T + ⊕T − , and abstract single and double layer operators constructed from G. This can indeed be achieved with the help of an appropriate quasi boundary triple for T which also permits one to introduce a natural abstract analog of the "boundary Hilbert space" L 2 (C) in the concrete case of the Laplacian above.
In Section 2 we briefly recall the basic setup for quasi boundary triples and associated operator-valued Weyl-functions (also called Weyl-Titchmarsh functions) as needed in this paper. The introduction of quasi boundary triples is intimately connected with an abstract (second) Green identity. Section 3 studies the operator A and derives Krein-type resolvent formulas for it in terms of A 0 and a related operator. Section 4 derives the abstract third Green identity, and finally Section 5 illustrates the abstract material in Sections 2-4 with the concrete case of Schrödinger operators on Lipschitz domains on smooth, boundaryless, compact Riemannian manifolds.
Finally, we briefly summarize the basic notation used in this paper: Let H, H be separable complex Hilbert spaces, (·, ·) H the scalar product in H (linear in the second factor), and I H the identity operator in H. If T is a linear operator mapping (a subspace of ) a Hilbert space into another, dom(T ) denotes the domain of T . The closure of a closable operator S is denoted by S. The spectrum and resolvent set of a closed linear operator in H will be denoted by σ(·) and ρ(·), respectively. The Banach space of bounded linear operators in H is denoted by B(H); in the context of two Hilbert spaces, H j , j = 1, 2, we use the analogous abbreviation B(H 1 , H 2 ). The set of all closed linear operators in H is denoted by C(H). Moreover, X 1 → X 2 denotes the continuous embedding of the Banach space X 1 into the Banach space X 2 . We also abbreviate C ± := {z ∈ C | Im(z) ≷ 0}.

Quasi Boundary Triples and their Weyl Functions
In this section we briefly recall the notion of quasi boundary triples and the associated (operator-valued) Weyl functions.
In the following let S be a densely defined closed symmetric operator in H.
is a Hilbert space and Γ 0 , Γ 1 : dom(T ) → H are linear mappings such that the following items (i)-(iii) hold: (i) The abstract (second ) Green identity is valid. The notion of quasi boundary triples was introduced in [5] and extends the concepts of ordinary (and generalized) boundary triples, see, for instance, [11], [19], [21], [27], and the references therein. We recall that the triple in Definition 2.1 is called an ordinary boundary triple (generalized boundary triple) if item (ii) is replaced by the condition ran(Γ) = H 2 (ran(Γ 0 ) = H, respectively). On the other hand, the notion of quasi boundary triple is a particular embodiment of the more general notion of isometric/unitary boundary triples. The latter goes back to Calkin [13] and was studied in detail in [15], [17].
We recall briefly some important properties of quasi boundary triples. First of all, we note that a quasi boundary triple for S * exists if and only if the defect numbers n ± (S) = dim(ker(S * ∓ i)) is automatically self-adjoint and the the quasi boundary triple {H, Γ 0 , Γ 1 } is an ordinary boundary triple in the usual sense. In this context we also note that in the case of finite deficiency indices of S a quasi boundary triple is automatically an ordinary boundary triple.
Next, the notion of the γ-field and Weyl function associated to a quasi boundary triple will be recalled. First, one observes that for each z ∈ ρ(A 0 ), the direct sum decomposition holds. Hence the restriction of the mapping Γ 0 to ker(T − zI H ) is injective and its range coincides with ran(Γ 0 ).

4)
and The notions of γ-field and Weyl function corresponding to ordinary and generalized boundary triples have been introduced in [18] and [19], respectively; the definition above is formally the same. In the special case of ordinary and generalized boundary triples the Weyl function M turns out to be a Herglotz-Nevanlinna function with values in B(H), that is, M is holomorphic on C\R, and Im(z) Im(M (z)) 0 and M (z) = M (z) * , z ∈ C\R. (2.6) The values of the γ-field are bounded operators from H into H with ran(γ(z)) = ker(T − zI H ) and the following identity holds In the case of a quasi boundary triple the operators γ(z), z ∈ ρ(A 0 ), are defined on the dense subspace ran(Γ 0 ) ⊆ H and map onto ker(T − zI H ) ⊂ H. By [5, Proposition 2.6] the operator γ(z) is bounded and hence admits a continuous extension onto H. Furthermore, one has The values of the Weyl function M (z), z ∈ ρ(A 0 ) are operators in H defined on ran(Γ 0 ) and mapping into ran(Γ 1 ). The analogs of (2.6) and (2.7), and various other useful and important properties of the Weyl function can be found in [5], [6]. In particular, 9) and hence the operators M (z), z ∈ ρ(A 0 ), are closable operators in H. We point out that the operators M (z), z ∈ ρ(A 0 ), and their closures are generally unbounded.

The Coupling Model
In this section we discuss the coupling issue mentioned in (1.15)-(1.17) from a purely abstract point of view.
Let S + and S − be densely defined closed symmetric operators in the separable Hilbert spaces H + and H − , respectively, and assume that the defect indices of S + and S − satisfy The case of finite defect numbers can be treated with the help of ordinary boundary triples in an efficient way and will not be discussed here (cf. [14]).
Let T + and T − be such that T + = S * + and T − = S * − , and assume that are quasi boundary triples for T + ⊆ S * + and T − ⊆ S * − , respectively. The corresponding γ-fields and Weyl functions are denoted by γ + and γ − , and M + and M − , respectively. Furthermore, let It is important to note that the identities hold for all z ∈ ρ(A 0,+ ) and z ∈ ρ(A 0,− ), respectively (cf. (2.8)). In the following consider the operators and that The elements f in the domain of S, T and S * will be written as two component vectors of the is a quasi boundary triple for T ⊆ S * such that The γ-field γ and Weyl function M corresponding to the quasi boundary triple {H ⊕ H, Γ 0 , Γ 1 } are given by and The next result, Theorem 3.1, can be viewed as an abstract analogue of the coupling of differential operators, where Γ ± 0 are Dirichlet trace operators and Γ ± 1 are Neumann trace operators acting on different domains (cf. Section 5 for more details). We also note that in the following the operators M + (z) + M − (z) in H are assumed to be defined on dom(M + (z)) ∩ dom(M − (z)).
holds for some (and hence for all ) z ∈ C + and some (and hence for all ) z ∈ C − .
If A is a self-adjoint operator in H then for all z ∈ ρ(A) ∩ ρ(A 0 ) the resolvent of A is given in terms of a Krein-type resolvent formula by Remark 3.2. The perturbation term γ(z)Θ(z)γ(z) * in the right-hand side of (3.15) may also be written in the form Proof of Theorem 3.1. (i) In order to show that A is a symmetric operator in H let f = (f + , f − ) and g = (g + , g − ) be in dom(A). Making use of the abstract boundary conditions for f, g ∈ dom(A), a straightforward computation using the abstract Green identity (2.1) shows that Conversely, assume that ϕ ∈ ker(M + (z) + M − (z)), ϕ = 0, for some z ∈ ρ(A 0 ). One notes that Hence there exist f + ∈ ker(T + − zI H+ ) and f − ∈ ker(T − − zI H− ) such that From the definition of M + and M − , and (3.21)-(3.22) one concludes that (iii) First, assume that A is a self-adjoint operator in H, fix z ∈ C\R, and let Then there exists f + ∈ H + such that where the last identity follows from (3.3). and Making use of (3.25) and f − = 0 this reads componentwise as (cf. (3.11)). Summing up these two equations and taking into account that k ∈ dom(A) satisfies Γ Hence, the inclusion holds for any z ∈ C\R. In the same way as above one also shows the inclusion Next, we will prove the converse. Assume that there exist z ± ∈ C ± such that holds for z ∈ {z ± }. We have to prove that the operator A is self-adjoint in H. Along the way we will also show that the resolvent formula holds at the point z.
Note first that A is symmetric by item (i) and hence all eigenvalues of A are real.
In particular, z is not an eigenvalue of A and according to item (ii), the operator and that 35) by assumption. Now consider the element 36) which is well-defined by the above considerations and the fact that and it is clear that g ∈ dom(T ). Next, it will be shown that g = (g + , g − ) satisfies the boundary conditions and the special form of γ(z) and γ(z) * , one infers that

41) and
(3.42) Since by definition, A 0,± = T ± ker(Γ ± 0 ) and γ ± (z) = (Γ ± 0 ker(T ± − zI H± )) −1 , one obtains and hence the first condition in (3.39) is satisfied. Next, we make use of (3.3) and M ± (z) = Γ ± 1 γ ± (z) and compute It follows that Γ + 1 g + + Γ − 1 g − = 0. Hence also the second boundary condition in (3.39) is satisfied. Therefore, g ∈ dom(A), and when applying (A − zI H ) to g it follows from the particular form of g and ran(γ(z)) ⊆ ker(T − zI H ) that Furthermore, as A is symmetric, z is not an eigenvalue of A and one concludes that Since f ∈ H was chosen arbitrary it follows that (A − zI H ) −1 is an everywhere defined operator in H. By our assumptions this is true for z = {z ± }. Hence it follows that A is self-adjoint and that the resolvent of A at the point z has the asserted form, proving assertion (iii).
It remains to show that the resolvent of A is of the form as stated in the theorem for all z ∈ ρ(A) ∩ ρ(A 0 ). For this we remark that The next step is to derive a slightly modified formula for the resolvent of A in Theorem 3.1 where the resolvent of A 0 is replaced by the resolvent of the operator ) is assumed to be a self-adjoint operator in H − . We recall that in the context of quasi boundary triples, the extension A 1,− of S − corresponding to ker(Γ − 1 ) is always symmetric, but generally not self-adjoint. The resolvent formula in the next theorem is essentially a consequence of Theorem 3.1 and a formula relating the resolvent of A 0,− with the resolvent of A 1,− .

The Third Green Identity
This section is devoted to an abstract version of the Third Green identity (cf. (1.19) for the concrete example that motivated these investigations).
Let T be given by (3.4) and let {H, Γ + 0 , Γ + 1 } and {H, Γ − 0 , Γ − 1 } be quasi boundary triples for T + ⊆ S * + and T − ⊆ S * − , respectively. We will investigate the operator which corresponds to the coupling of the quasi-boundary triples {H, Γ + 0 , Γ + 1 } and {H, Γ − 0 , Γ − 1 } in Theorem 3.1. One recalls that A is a symmetric operator in the Hilbert space H = H + ⊕ H − . From now on we shall assume that the following hypothesis is satisfied (cf. Theorem 3.1 (iii)).   Since A ∈ B(H 2 , H), also A ∈ B(H, H −2 ). Next, define the map Υ on H 2 = dom(A) as the restriction of (Γ + 0 , Γ + 1 ) , Here the last equality follows from the abstract boundary conditions in (4.1) for all f ∈ dom(A). Since Υ j are bounded operators from H 2 to H by Lemma 4.2, it is clear that Υ * j , j = 1, 2, are bounded operators from H to H −2 .
Next we introduce an abstract analog of the single and double layer potential (cf. [29]). For this it will be assumed that there is an abstract fundamental solution operator for A.   respectively.
It is clear that the operators S and D are well-defined. In order to obtain an abstract third Green identity in the next theorem we will also use the following notations for the "jumps" of boundary values: and [ (4.15) One observes that the jump notations in (4.14)-(4.15) are compatible with the boundary conditions for elements in dom(A); we note that different signs are used in (4.15) since in the application in Section 5 the operators Γ ± 1 will be the normal derivatives with the normals having opposite direction. Proof. Let f = (f + , f − ) ∈ dom(T ) and g = (g + , g − ) ∈ dom(A). Then it follows from (4.6) and the abstract Green identity (2.1) that (4.17) As g ∈ dom(A) one concludes that by (4.7), and hence (4.17) takes on the form where (4.14)-(4.15) were used in the second equality, and (4.10) was employed in the last equality. Since (4.19) is true for all g ∈ dom(A) = H 2 one concludes that (4.20) and making use of the definition of G and (4.12)-(4.13) one finally obtains as desired.
The following corollary may be viewed as an abstract unique continuation result.

Coupling of Schrödinger Operators on Lipschitz Domains on Manifolds
In this section we illustrate the abstract material in Sections 2-4 with the concrete case of Schrödinger operators on Lipschitz domains on boundaryless Riemannian manifolds, freely borrowing results from [4]. For more details and background information concerning differential geometry and partial differential equations on manifolds the interested reader is referred to [30], [31], [37], and the literature cited there.
Suppose (M, g) is a compact, connected, C ∞ , boundaryless Riemannian manifold of (real) dimension n ∈ N. In local coordinates, the metric tensor g is expressed by As is customary, we shall use the symbol g to also abbreviate g := det (g jk ) 1≤j,k≤n , (5.2) and we shall use (g jk ) 1≤j,k≤n to denote the inverse of the matrix (g jk ) 1≤j,k≤n , that is, (g jk ) 1≤j,k≤n := (g jk ) 1≤j,k≤n −1 .
3) The volume element dV g on M with respect to the Riemannian metric g in (5.1) then can be written in local coordinates as Following a common practice, we use {∂ j } 1≤j≤n to denote a local basis in the tangent bundle T M of the manifold M . This implies that if X, Y ∈ T M are locally expressed as X = n j=1 X j ∂ j and Y = n j=1 Y j ∂ j , then where ·, · T M stands for the pointwise inner product in T M . Next, we discuss the gradient and divergence operators associated with the metric g on the manifold M . Specifically, given an open set Ω ⊂ M and some function f ∈ C 1 (Ω), the gradient of f is the vector field locally defined as (5.6) Also, given any vector field X ∈ C 1 (Ω, T M ) locally written as X = n j=1 X j ∂ j , its divergence is given by div g (X) := where Γ i jk are the Christoffel symbols associated with the metric (5.1). The Laplace-Beltrami operator ∆ g := div g grad g , (5.8) is expressed locally as We are interested in working with the Schrödinger operator and H 0 (∂Ω) coincides with L 2 (∂Ω), the space of square-integrable functions with respect to the surface measure σ g induced by the ambient Riemannian metric on ∂Ω. Moreover, and H s0 (∂Ω) → H s1 (∂Ω) continuously, whenever − 1 ≤ s 1 ≤ s 0 ≤ 1.
In the following functions on M will be identified with the pair of restrictions onto Ω + and Ω − and a vector notation will be used. For example, for f ∈ L 2 (M ) we shall also write (f + , f − ) , where f ± ∈ L 2 (Ω ± ). This notation is in accordance with the notation in Sections 3 and 4. Also, in the sequel we agree to abbreviate For s ≥ 0 we define the Banach spaces The minimal and maximal realizations of −∆ g + V ± in L 2 (Ω ± ) are defined by S min,± := −∆ g + V ± , dom(S min,± ) =H 2 (Ω ± ), (5.23) and S max,± := −∆ g + V ± , dom(S max,± ) = H 0 ∆ (Ω ± ). (5.24) In the next lemma we collect some well-known properties of the operators S min,± and S max,± . A proof of this lemma and some further properties of the minimal and maximal realization of −∆ g + V ± can be found, for instance, in [4].
Lemma 5.1. The operators S min,± and S max,± are densely defined and closed in L 2 (Ω ± ). The operator S min,± is symmetric, semibounded from below, and has infinite deficiency indices. Furthermore, S min,± and S max,± are adjoints of each other, that is, S min,± * = S max,± and S min,± = S max,± * . (5.25) Let n ± ∈ L ∞ (C, T M ) be the outward unit normal vectors to Ω ± . One observes that in the present situation n + = −n − . The Dirichlet and Neumann trace operators τ ± D and τ ± N , originally defined by τ ± D f ± := f ± C , τ ± N f ± := n ± , grad g f ± C T M (5.26) for f ± ∈ C ∞ (Ω ± ), admit continuous linear extensions to operators whose actions are compatible with one another, for all s ∈ [ 1 2 , 3 2 ]. We refer to [4] where it is also shown that the trace operators τ ± D , τ ± N in (5.27) are both surjective for each s ∈ [ 1 2 , 3 2 ]. (5.28) We wish to augment (5.28) with the following density result.
Lemma 5.2. The ranges of the mappings are dense in L 2 (C) × L 2 (C).
Let us also consider their boundary versions, that is, the singular integral operators acting on an arbitrary function ϕ ∈ L 2 (C) according to and where P.V. indicates that the integral is considered in the principal value sense (i.e., removing a small geodesic ball centered at the singularity and passing to the limit as its radius shrinks to zero). Work in [33]- [35] ensures that the following properties hold: τ ± D (S ± 0 ϕ) = S ± 0 ϕ on C, for each function ϕ ∈ L 2 (C), (5.45) τ ± N (S ± 0 ϕ) = − 1 2 I + (K ± 0 ) * ϕ on C, for every ϕ ∈ L 2 (C), (5.46) where I is the identity operator on L 2 (C), and (K ± 0 ) * are the adjoints of the operators K ± 0 in (5.40). After this preamble, we are ready to present the proof of Lemma 5.2.
Proof of Lemma 5.2. The density results claimed in the statement follow as soon as we establish that whenever two functions h D , h N ∈ L 2 (C) satisfy then necessarily h D = 0 and h N = 0. To this end, pick an arbitrary h ∈ L 2 (C) and consider f ± := S ± 0 h in Ω ± . Then f ± ∈ H 3/2 ∆ (Ω ± ) due to (5.42). Also, relying on (5.43) and the fact that, by design, the potentials V ± 0 vanish in Ω ± , we may write Granted these properties of f ± , from (5.47), (5.45), (5.46), and (5.44), one concludes that With this in hand, the arbitrariness of h ∈ L 2 (C) then forces Next, we pick two arbitrary functions φ ± ∈ C ∞ 0 (Ω ± ) and, this time, consider f ± := G ± 0 φ ± in Ω ± . Then (5.32) ensures that f ± ∈ H 2 (Ω ± ). Given that, by design, Having established these properties of f ± , (5.47) implies that Now we take a closer look at the two terms in the left-hand side of (5.51). For the first term we write where the first equality uses the definition of E ± 0 (x, y), the second equality is based on Fubini's theorem, while the third equality is a consequence of (5.33) and (5.34). For the second term in (5.51) we compute where the first equality relies on the definition of E ± 0 (x, y), the second equality uses Fubini's theorem, while the third equality is implied by (5.33) and (5.35).
Together, (5.51), (5.52), and (5.53) imply that Applying τ ± D to both sides of (5.55) then yields, on account of (5.45) and (5.41), The end-game in the proof of the lemma is as follows. Subtracting (5.56) from (5.50) proves that h N = 0. Using this back into (5.56) leads to S ± 0 h D = 0 which, in light of the injectivity of the single layer operators in (5.44), shows that h D = 0 as well. Hence, h D = h N = 0, as desired.
Going further, in the next theorem we define quasi boundary triples for S max,± = S min,± * with the natural trace maps as boundary maps defined on the domain of the operators (5.57) in L 2 (Ω ± ). One recalls that With this choice of boundary maps the values of the corresponding Weyl function are Dirichlet-to-Neumann maps (up to a minus sign).
Proof. Let us verify the properties stipulated in Definition 2.1 in the current case. First, the abstract Green identity (2.1) presently corresponds to the second Green identity for the Schrödinger operator (5.10) on the Lipschitz domain Ω, proved in [4]. Second, the fact that ran(Γ ± 0 , Γ ± 1 ) is dense in L 2 (C) × L 2 (C) is readily implied by Lemma 5.2 (bearing in mind (5.21)). Third, the self-adjointness of A 0,± = T ± ker(Γ ± 0 ) is clear from the fact that these operators coincide with the self-adjoint Dirichlet realizations of −∆ g + V ± in Ω ± studied in [4].
Fourth, we focus on establishing that T ± = S max,± . In turn, since dom(T ± ) contains dom(A D,± ) + dom(A N,± ), this is going to be a consequence of the fact that dom(A D,± ) + dom(A N,± ) is dense in dom(S max,± ) with respect to the graph norm. (5.66) To prove (5.66), we assume that h ∈ dom(S max,± ) is such that one concludes that h = 0. Hence (5.66) holds, completing the proof of the fact that T ± = S max,± . This shows that {L 2 (C), Γ ± 0 , Γ ± 1 } are indeed quasi boundary triples for T ± . That T ± ⊂ S max,± is clear from definitions, while (5.60) has been established in [4].
Thanks to work in [4], the assertions in (ii) and (iii) follow immediately from the definition of the γ-field and the Weyl function. We refer the interested reader to [4] for more details. Here we only wish to note that in the case of a bounded Lipschitz domain in the flat Euclidean setting (i.e., R n equipped with the standard metric) a similar result has been established in [8,Theorem 4.1].
In the following we establish the link to the coupling procedure discussed in Section 3. First of all we set H := L 2 (C) so that the quasi boundary triples in Section 3 are those in Theorem 5.3. The operator S in (3.4) is the direct orthogonal sum of the minimal realizations S min,+ and S min,− , and the boundary mappings in the quasi boundary triple {L 2 (C) ⊕ L 2 (C), Γ 0 , Γ 1 } in (3.7) are now given by where f = (f + , f − ) ∈ dom(T ) with T ± , T given as in (5.57), (5.58). The self-adjoint operator corresponding to ker(Γ 0 ) is the orthogonal sum of the Dirichlet operators A D,+ and A D,− in L 2 (Ω + ) and L 2 (Ω − ), respectively, The following lemma shows that the coupling of the quasi boundary triples {L 2 (C), Γ + 0 , Γ + 1 } and {L 2 (C), Γ − 0 , Γ − 1 } in Theorem 3.1 leads to the self-adjoint Schrödinger operator in (5.16).
Lemma 5.4. The operator coincides with the self-adjoint operator A in (5.16). Proof.
Since any function f ∈ H 2 (M ) satisfies and f ± ∈ H 2 (Ω ± ) ⊂ H 3/2 ∆ (Ω ± ) = dom(T ± ), it follows that H 2 (M ) = dom(A) is contained in the domain of the operator in (5.75). On the other hand, it follows from Theorem 3.1 (i) that the operator in (5.75) is symmetric, and hence self-adjoint (as it extends the self-adjoint operator A).
As an immediate consequence of the observation in Lemma 5.4 we obtain the next corollary. First, we note that the self-adjointness of the operator A in (5.16), Theorem 3.1, and the fact that where M ± are (minus) the Dirichlet-to-Neumann maps in (5.65).
Corollary 5.5. Let A 0 be the orthogonal sum of the Dirichlet operators in (5.74), let M ± be the (minus) Dirichlet-to-Neumann maps in (5.65) and let γ be the orthogonal sum of the γ-fields in (5.63) (cf. (3.9)). For all z ∈ ρ(A) ∩ ρ(A 0 ), the resolvent of A is given by In a similar way one obtains a representation for the resolvent of A from Theorem 3.3, where (A − zI L 2 (M ) ) −1 is compared with the orthogonal sum of the selfadjoint Dirichlet operator A D,+ in L 2 (Ω + ) and the self-adjoint Neumann operator A N,− in L 2 (Ω − ). which are in agreement with (5.34), (5.35). Furthermore, the abstract jump relations in (4.14)-(4.15) are [ As a consequence of the above considerations and Theorem 4.5 we obtain the following version of Green's third identity for the Schrödinger operator −∆ g + V on M . ∆ (Ω ± )). In conclusion, we note that boundary triples for elliptic operators in an unbounded external domain Ω − ⊂ R n , used as an illustration in the introduction, were studied, for instance, in [7], [28]. The third Green formula in this situation and its analog in connection with noncompact Riemannian manifolds M requires additional techniques to be discussed elsewhere.