Generalized boundary triples, II. Some applications of generalized boundary triples and form domain invariant Nevanlinna functions

The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$ ‐generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.

To characterize the class of -generalized boundary triples in terms of the corresponding Weyl functions we associate with each (⋅) ∈ () a family of nonnegative quadratic forms ′ ( ) in : According to [26,Theorem 1.14], Weyl functions (⋅) corresponding to -generalized boundary triples are characterized by the following form domain invariance property: ′ ( ) is closable for each ∈ ℂ ± and the domain of its closure ( ) ∶= ′ ( ) 1 , called the form domain of the Weyl function (⋅), does not depend on ∈ ℂ ± . In this case dom ( ) may, or may not, depend on ∈ ℂ ⧵ ℝ, while for -generalized boundary triples the equality dom ( ) = ran Γ 0 follows from the decomposition * = 0+ˆi n Theorem 1.3 (ii) and hence the Weyl function is always domain invariant and then also form domain invariant by [26,Proposition 5.30], see also [24] for general invariance results on operator-valued Nevanlinna functions. For an -generalized, but not -generalized, boundary triple 0 is only essentially selfadjoint and then one has just strict inclusions * ⫋ 0+ˆ. Therefore, the equality dom ( ) = ran Γ 0 is violated and a strict inclusion dom ( ) ⫋ ran Γ 0 holds; cf. discussions following [26,Theorems 1.13,1.14]. Another characteristic difference between -generalized and -generalized boundary triples appears in the -field: by Theorem 1.3 (iv) for -generalized boundary triple ( ) is a bounded operator for all ∈ ℂ ⧵ ℝ, while for -generalized boundary triple ( ) is, in general, an unbounded operator, which is closable for all ∈ ℂ ⧵ ℝ. Furthermore, as shown in [26,Theorem 5.24] for generalized boundary triples the form domain of the Weyl function (⋅) is directly connected to the closures of ( ) and Γ 0 ∶ * →  by the following characteristic identities: dom ( ) = dom ( ) = ran Γ 0 .
These facts will be demonstrated in concrete boundary value problems: for Laplace operators on smooth domains in Theorem 3.1, where the form domain of the Weyl function associated with the Kreȋn-von Neumann Laplacian is described explicitly; see (3.13). Similarly it is shown that -generalized boundary triples occur naturally when describing mathematical models for various physical phenomena involving Schrödinger, Dirac, and momentum operators with local point interactions; cf. [3,35]. In particular, in Proposition 4.8 such an -generalized boundary triple occurs in the connection with momentum operators. It is shown therein that the domain and the form domain of the associated Weyl function (⋅) admit the following explicit descriptions: see (4.13), (4.14). Here = { } ∞ 1 ⊂ ℝ + is a strictly increasing sequence of point interactions satisfying two conditions lim →∞ = ∞ and * ∶= inf ∈ℕ = 0; here = − −1 > 0. Similar explicit descriptions for the domain and form domain of the function ( ) are also presented for local point interactions involving Schrödinger operators in Theorem 4.10, see formulas (4.19)- (4.21), as well as in the case of Dirac operators in Proposition 4.19; see (4.48), (4.49).
Here is a short description of the contents of Part II. Section 2 contains a couple of further useful results which are of preparatory nature for applications of unitary and, in particular, -generalized boundary triples. Namely, it is shown how certain simple transforms of -generalized boundary triples generate -generalized boundary triples; see Theorems 2.1, 2.2. On the other hand, by applying some proper renormalization procedures for -generalized boundary triples one can produce more regularly behaving boundary mappings; cf. Theorem 2.6. The key to find appropriate kind of transforms and renormalization procedures is based on the behaviour of the corresponding Weyl functions under such transforms, and hence these constructions are basically motivated by the analytic properties of the associated Weyl functions. The connection of various classes of boundary pairs for nonnegative forms as defined in Post [62] to various subclasses of generalized boundary triples is established in Theorem 2. 16. For instance, we show that the so-called elliptically regular boundary pair as introduced in [62] generates an -generalized boundary triple with a nonnegative operator and vice versa. Section 3 is devoted to applications of the general results in the PDE setting by treating Laplace operators in smooth bounded domain in Theorem 3.1 and for Lipschitz domains in Proposition 3.7. Mixed boundary value problems for Laplacian are also considered and again an -generalized boundary triple occurs in the connection of so-called Zaremba Laplacian; see Theorem 3.5. Laplacian on rough domains is shown to lead to a multivalued boundary mapping Γ and its multivalued transposed mapping Γ ⊤ (called here unitary boundary pairs) where the corresponding Weyl function can even be multivalued; see Theorem 3.12. In Section 4 spectral problems for momentum, Schrödinger and Dirac operators with local point interactions are treated from the point of view of boundary triples technique. The new subclasses of generalized boundary triples from Part I and the corresponding analytic properties of associated Weyl functions allow to complete the results of [22,52,53], and [58]. In particular, it is shown, see Proposition 4.8, Theorem 4.10, and Proposition 4.19, that in each of these three cases the Weyl function is domain invariant and form domain invariant and we describe explicitly all of these domains. In these applications to local point interactions the underlying abstract results become demonstrated in a concrete way and the obtained results simultaneously allow, for instance, a straightforward verification of the specific properties of the corresponding Weyl functions associated with the different types of generalized boundary triples occurring therein.
We devote this paper to our dear friend and excellent mathematician Hagen Neidhardt who passed away in March, 2019. One of us collaborated with Hagen a lot in applications of boundary triples technique to the spectral and scattering theory. It is a great loss for us as well as for the whole spectral theory community.
The situation changes essentially when { , Γ 0 , Γ 1 } is not an ordinary boundary triple for * . In this section we treat the simplest case of a -generalized boundary triple and show that a simple  -unitary transform can produce a boundary triple for * which is not -generalized and not even -generalized. More precisely, the next result shows how any -generalized boundary triple { , Γ 0 , Γ 1 } for * , which is not an ordinary boundary triple, can be transformed to an -generalized boundary triple, whose -field becomes unbounded.
be a -generalized boundary triple for * with * = dom Γ ⊂ * , * ≠ * , let (⋅) and (⋅) be the corresponding Weyl function and -field, and let 0 = ker Γ 0 . Then: (i) for every fixed ∈ ℂ ⧵ ℝ the transform defines a unitary boundary triple for * whose Weyl function and -field are given by Proof.
□ Theorem 2.1 will now be specialized to a situation that appears often in system theory and in PDE setting where typically the underlying minimal symmetric operator is nonnegative; the simplest situation occurs when the lower bound is positive. The first part of the next result follows the general formulation given in [30,Proposition 7.41] which was motivated by the papers of V. Ryzhov; see [63] and the references therein. Theorem 2.2. Let 0 be a selfadjoint relation in ℌ with ker 0 = {0}, let be a selfadjoint operator in , and let the operator ∶  → ℌ be bounded and everywhere defined with ker = {0}. Moreover, let * = and define the operators Γ 0 , Γ 1 ∶ * →  by Then: and ∈ dom the corresponding -field and the Weyl function are given by and whose Weyl function and -field are given bỹ and, moreover, the transposed boundary tripleΠ ⊤ is -generalized with the Weyl function 0 (⋅);

8)
where stands for the orthogonal projection onto ran . Proof.
(i) It was proved in [30,Prop. 7.41] that Π is a unitary boundary triple for * = * and for (i) it suffices to note that ker Γ 0 = 0 is selfadjoint by assumption. Hence, Π is an -generalized boundary triple.
The formula for Γ 0 shows that ran Γ 0 =  precisely when dom =  or equivalently, is bounded. Since and in the last product the triangular operator is bounded with bounded inverse when is bounded, we conclude that ran Γ =  ×  if and only if dom =  and the diagonal operator in (2.9) is surjective, i.e., * ( ran 0 ) = ; in this case ran * =  and ran is closed. (iv) It is clear from (2.5) that the transform { Γ 1 − Γ 0 , −Γ 0 } has the same domain * as Γ. Moreover, using (2.5) it is straightforward to check that the closure is given by (2.6). In fact, the transposed boundary triple is -generalized and of the same form as Γ in (2.5) when = 0, i.e., in view of (ii) it is even -generalized. Applying (i) to this transposed boundary triple one also concludes that the Weyl function and -field of the boundary triple {Γ 0 ,Γ 1 } are given by (2.7). (v) It follows from (2.6) that˜0 = kerΓ 0 is given bỹ (2.10) Using graph expressions one can write˜0 = 0 ∩ ( × ker * )+ (ran × {0}) and now using the properties of adjoints it is seen that˜ * 0 = clos then 0+ ran × {0} is a closed subspace of ℌ 2 and this implies that˜ * 0 =˜0. Hence,˜0 is essentially selfadjoint. Since 0 ∈ ( 0 ) , it is clear from (2.10) that ran is closed if and only if˜0 = kerΓ 0 is closed, or equivalently, domΓ in (2.6) is closed. (vi) Using for˜0 the formula (2.10) and the equalities ( domΓ ) = ran˜( ) = ran ( ) the domain invariance condition in [26,Proposition 3.11] can be rewritten as follows: for every ℎ ∈  there exist ℎ 0 ∈  and ′ ∈ ran 0 ∩ ker * such that Applying resolvent identity to the product term it is seen that the previous condition is equivalent to for some ℎ 1 ∈  and ′ 1 ∈ ran 0 ∩ ker * . This condition is equivalent to the inclusion Since , ∈ ℂ ⧵ ℝ are arbitrary, this last condition coincides with the condition (2.8). □ Remark 2.3.
(ii) If is bounded, no closure is needed in part (iv), i.e.,Γ = { Γ 1 − Γ 0 , −Γ 0 } . In this case, Γ is a -generalized boundary triple and Theorem 2.2 can be seen as an extension of Theorem 2.1 to a point on the real line. Here the results are formulated for = 0. They can easily be reformulated also for ∈ ℝ. In addition, for = ∞ the results in Theorem 2.2 can be translated to analogous results for range perturbations (instead of domain perturbations as in Theorem 2.2); for general background see [30,Section 7.5]. For = ∞ the operator appears as the limit value (∞), while 0 and * should be replaced by their inverses; see (2.15). (iii) The criterion (2.8) for domain invariance of˜can be derived also directly using dom˜( ) = ran 0 ( ) and the explicit formula for 0 ( ) given in part (iv) of Theorem 2.2; see also the equivalent condition in (2.11).
is not an ordinary boundary triple for * , the condition (2.8) fails to hold in general. In particular, if ran 0 ∩ ker * = {0} (if e.g. ker * = {0}), then the condition (2.8) is equivalent to If is densely defined, then * ⊃ * is an operator. Since ker 0 = {0} one concludes from (2.4) that * is an operator if and only if dom 0 ∩ ran = {0}. This condition applied to (2.14) implies that = 0 and ℎ − = 0, since ker = {0}. This proves the claim. □ If 0 in Theorem 2.2 is nonnegative, one can specify further the type of the Weyl function as follows. . This generates the following expression for an associated Schur complement of the resolvent  [26,Theorem 5.32], is an ordinary boundary triple for * = 0+ (ran × {0}) and is determined by where denotes the orthogonal projection onto ran ; (iii) the Weyl function (⋅) of Π coincides with the Schur complement in (2.16), , and the form domain invariant Weyl function˜(⋅) in (2.7) has the form where ( * ) is the adjoint when is treated as an operator from  into ran . Proof.
(i) By (v) Theorem 2.2˜( ) = ( ) ( 0 ( ) ) −1 . Using the expressions for 0 ( ) in (2.7) and 0 ( ) in (2.16) one obtains where −( * ) stands for the inverse of * , when * is treated as an injective mapping from ran to . Since , we conclude that the form domain of˜( ) is equal to ran * and that the closure of the -field is given bỹ . Here the last identity uses the standard block formula for the inverse implies that the closure of domΓ is * = 0+ (ran × {0}). In view of (i) one can use * ∶ ran →  as the renormalizing operator in [26,Theorem 5.32]. Now in view of expression forΓ in (2.6) this renormalization gives the formula The final expression for the renormalized boundary triple Π is obtained by taking closure in (2.20); this leads to (2.17), since 0 ∈ ( 0 ) . Now clearly dom Γ = * and ran Γ = ran × ran , i.e., Π is an ordinary boundary triple for * . (iii) This follows from (2.19). For the equality ( ) = 0 ( ) take the closure of˜( ) * ↾ ran .

□
According to Theorem 2.6 0, = ker Γ 0, is selfadjoint. Clearly, 0, coincides with the closure of˜0 = kerΓ 0 in Theorem 2.2; see (2.10). If, in particular, 0 is strictly positive, then 0, = ker Γ 0, is the Kreȋn-von Neumann extension of and we have the following identities where is the range restriction of 0 :  (2.5). Then ker Γ 0 = 0 is the unperturbed relation and In particular, if 0 is an operator then is a range restriction of 0 to ker * with ± ( ) = dim (ran ). Now, assume that ker = {0} and ran * ∩ ran = {0}. Then the identity * ′ + = 0 implies that * ′ = = 0 and, consequently, = 0 and this means that 1 = . This means that 1 is not essentially selfadjoint and thus the transposed boundary triple { , Γ 1 , −Γ 0 } is not -generalized. The corresponding Weyl function is given by and according to [26,Theorem 1.14] it cannot be form domain invariant.
If, in addition, ker * = {0}, then To see this assume that holds for some , 1 , 2 ∈ . Then Finally, it should be mentioned that later, in Section 3, it is shown how the standard Dirichlet and Neumann trace operators on smooth, as well as on Lipschitz, domains can be included in the abstract boundary triple framework constructed in Theorem 2.2; hence the previous results can be made explicit in PDE setting.

Graph continuity of boundary mappings
It is known that for a boundary triple { , Γ 0 , Γ 1 } (as well as for a boundary pair {, Γ}, see [26, Definition 3.1]) to be an ordinary boundary triple it is necessary and sufficient that both boundary mappings Γ 0 and Γ 1 are continuous on * (with the graph norm on dom * in case is densely defined). In general the mappings Γ 0 and Γ 1 both can be unbounded when dim  = ∞. In this section we establish analytic criteria for Γ 0 or Γ 1 to be continuous with the aid of the associated Weyl function. Recall that the kernels 0 = ker Γ 0 and 1 = ker Γ 1 are always symmetric and it is possible that 0 = or 1 = ; see e.g. Example 2.7. The next result characterizes boundedness of the mapping Γ 1 for an -generalized boundary triple. (i) 0 = ker Γ 0 is essentially selfadjoint and Γ 1 is a bounded operator (w.r.t. the graph norm) on * ; (ii) 0 is selfadjoint and the restriction Γ 1 ↾ˆ( * ) is a bounded operator for some (equivalently for every) ∈ ℂ ⧵ ℝ; (iii) the form associated with Im ( − −1 ( ) ) has a positive lower bound for some (equivalently for every) ∈ ℂ ⧵ ℝ.

If one of the conditions is satisfied, then the triple
Proof.

If one of these conditions is satisfied, then the transposed boundary triple
Remark 2.10.
(i) For infinite direct sums of ordinary boundary triples the extensions = ker Γ , = 1, 2, are automatically essen- is a -generalized boundary triple for * by Proposition 2.8; this implication was proved in another way in [52, Proposition 3.6]; see also Corollary 4.6 below.
is a -generalized boundary triple if and only if the composition Γ 1ˆ( ) (= ( )) is bounded for some (equivalently for all) ∈ ℂ ⧵ ℝ. In particular, if Γ 1ˆ( ) is bounded, then also the -field ( ) itself is bounded (see [26,Equation (3.6)]), 0 = * 0 (by Theorem 1.3) and the restriction Γ 1 ↾ 0 is also bounded (by [26,Corollary 5.6]). However, in this case Γ 1 need not be bounded. Therefore, the conditions in Proposition 2.8 are sufficient, but not necessary, for Π to be a -generalized boundary triple. An example is any -generalized boundary triple Π, which is not an ordinary boundary triple, such that also the transposed boundary triple Π ⊤ is -generalized, since then Π ⊤ cannot be an ordinary boundary triple. Then the condition in (iii) of Proposition 2.8 is not satisfied.
For an explicit example of such a -generalized boundary triple, see local point interactions of Dirac operators treated in Proposition 4.17. Also the -generalized boundary triple for the Laplace operator associated with the Dirichletto-Neumann map in Theorem 3.1 (i) does not satisfy the properties in Proposition 2.9, but the transposed boundary triple is -generalized. On the other hand, the -generalized boundary triple associated with the Kreȋn-von Neumann Laplacian in Theorem 3.1 (ii) satisfies the conditions in Proposition 2.9 and the transposed boundary triple therein is -generalized.
The boundedness of the component mappings Γ 0 and Γ 1 can be used to derive the following new characterization of ordinary boundary triples.

Proposition 2.11. For a unitary boundary triple
Proof.

Extrapolation of Weyl functions via a real regular point
The main result here contains an analytic extrapolation principle for Weyl functions in the case when the underlying minimal operator admits a regular type point on the real line ℝ. The proof relies on the so-called main transform of boundary relations (called here boundary pairs) introduced in [28]. The main transform makes a connection between subspaces of the Hilbert space ( ℌ ⊕  ) 2 and linear relations from the Kreȋn space It is a linear mapping  from ℌ 2 ×  2 to (ℌ ⊕ ) 2 which establishes a one-to-one correspondence between all (closed) linear relations Γ ∶ ℌ 2 →  2 and all (closed) linear relations˜inH = ℌ ⊕  via According to [28, Proposition 2.10]  establishes a one-to-one correspondence between the sets of contractive, isometric, and unitary relations Γ from ( and the sets of dissipative, symmetric, and selfadjoint relations˜in ℌ ⊕ , respectively. Recall that a boundary pair {, Γ} is called minimal, if The next result shows usefulness of the main transform for analytic extrapolation of Weyl functions (⋅) from a single real point ∈ ℝ to the complex plane, when ∈̂( ) is a regular type point of the minimal operator . In the special case when the analytic extrapolation of ( ) is a uniformly strict Nevanlinna function the extrapolation principle formulated for Weyl functions in the next theorem, yields a solution to the following general inverse problem: given a pair { Γ 0 , Γ 1 } of boundary mappings from * to  determine the selfadjoint extension Θ of (up to unitary equivalence) when the boundary condition Γ 1ˆ= ΘΓ 0ˆi s fixed by some operator Θ acting on . It is emphasized that for this result it suffices to know initially only the value of ( ) at the single point ∈̂( ). In this case the value ( ) is defined in the same way as ( ) is defined for ∈ ℂ ⧵ ℝ (see [26,Definition 3.2]): ) or, more precisely,  (2.24). Then the following assertions hold: (i) The following two conditions are equivalent: are, up to unitary equivalence, uniquely determined by (⋅). Proof.
(ii) The proof of (i) shows that if (a) or, equivalently, (b) holds then˜is a selfadjoint relation in ℌ ⊕ . Thus the (inverse) main transform Γ = −1 (˜) defines is a unitary boundary pair {Γ, } for * . By the main realization result in [28, Theorem 3.9] one concludes that ∈(). (iii) To prove this assertion first recall that according to [28,Theorem 3.9] the Weyl function uniquely determines Γ, as well as˜, by the minimality of Γ. Uniqueness of Γ here means that if there exists another minimal boundary pair { ,Γ } associated with the symmetric operatorˆ= ker Γ in some Hilbert spaceĤ having the same Weyl function (⋅), then there exists a standard unitary operator ∶ ℌ →Ĥ such that , Hence, if the extension Θ of in the Hilbert space ℌ and the extensionˆΘ ofˆin the Hilbert spaceĤ are associated with the same "boundary condition" Θ then (2.27) implies that Thus Θ andˆΘ are unitarily equivalent via the same unitary operator for every linear relation Θ in . □ Remark 2.13. The proof of item (i) in Theorem 2.12 shows that (b) ⇒ (a) without the assumption on the existence of a selfadjoint extension ⊂ * with ∈ ( ). As to item (iii) of Theorem 2.12 it should be mentioned that if the analytic extrapolation (⋅) belongs to the class  [], then each selfadjoint extension Θ (Θ = Θ * ) of is uniquely (up to the unitary equivalence) defined by the Weyl function as well as by the (non-orthogonal) spectral measure Σ( ) from the integral representation of Θ (⋅), see [20,27,29,32,33] for details.
Some further developments concerning uniqueness of boundary triples and connections between ( Θ ) and the spectral functions Σ( ) can be found in [42]. Theorem 2.12 offers also a useful analytic tool to check whether an isometric boundary triple (or boundary pair) is actually unitary or, equivalently, if the Weyl function of some isometric boundary triple is in fact from the class () of Nevanlinna functions. We use this result to construct a unitary boundary pair for Laplacians defined on rough domains in Section 3.4 and to associate unitary boundary triples with boundary pairs of nonnegative forms in the next subsection.

Boundary pairs of nonnegative operators and boundary triples
The notion of boundary pairs involves initially only one boundary map associated with a closed nonnegative form or a pair of nonnegative selfadjoint operators. The purpose in this section is to show that, after introducing a second boundary map Γ 1 (via the first Green's identity), the boundary pair ( ,Γ 0 ) generates a unitary boundary triple { , Γ 0 , Γ 1 } . Furthermore, various special cases of boundary pairs are connected to specific classes of unitary boundary triples. In applications to PDE's is often the Neumann form and in abstract setting the form associated to the Kreȋn extension , which is the smallest nonnegative selfadjoint extension of . The notion of a boundary pair can be seen to arise from the works of Kreȋn, Birman, and Višik and has been treated in later papers by G. Grubb [7,50]. Boundary pairs which lead to -generalized boundary triples appear in [9]. A more general class of boundary pairs ( ,Γ 0 ) has been studied recently by O. Post [62]; who relaxed the surjectivity condition onΓ 0 and replaced it by the weaker requirement that ranΓ 0 is dense in . We recall the definition more explicitly here (using present notations): Definition 2.14 [62]. Let be a closed nonnegative form on a Hilbert space ℌ and letΓ 0 be a bounded linear map from is said to be a boundary pair associated with the form , if: is said to be bounded if ranΓ 0 = , otherwise it is said to be unbounded.
SinceΓ 0 is bounded its kernel defines a closed restriction of the form , which we denote here by 0 ( ) = ( ), ∈ kerΓ 0 . By assumption (a) the forms 0 and are densely defined in ℌ and we denote by 0 and the nonnegative selfadjoint operators associated with the closed forms 0 and , respectively. Next we associate a symmetric operator and its adjoint with the boundary pair .
In what follows we assume that is densely defined. By definition 0 and are disjoint selfadjoint extensions of .
, and there is similar decomposition with . Since 0 ⊂ , one and since clearly ran This sum is not in general direct, since − ∩ dom 0 is nontrivial, whenever 0 ≠ . Formulas (2.29) and (2.28) go back to the classical papers by Kreȋn [54] and Birman [19], respectively. Boundary triples approach to (2.28) as well as its further development including the case of operators with zero lower bound can be found in [56] (see also [43], [ leads to the following direct sum decomposition for every ∈ ( 0 ) : is closed in ℌ 1 , 1 is dense in ker( * − ), and 1 ∩ dom 0 = {0}. The restrictionΓ 0 ↾ 1 is a bounded operator from 1 into  and the decomposition (2.31) implies that it is injective and its range is equal to ranΓ 0 . The inverse operator is closed as an operator from  to ℌ 1 with domain  1∕2 = ranΓ 0 .
Definition 2.15 [62]. The boundary pair ( ,Γ 0 ) associated with the form is said to be elliptically regular, if the operator ∶= (−1) is bounded as an operator from  to ℌ, i.e. ‖ ℎ‖ ℌ ≤ ‖ℎ‖  for all ℎ ∈  1∕2 and some ≥ 0. Moreover, the boundary pair The form is closed in , since ∶  → ℌ 1 is a closed operator. By the first representation theorem, see [49], there is a unique selfadjoint operator Λ in  characterized by the equality It is clear that Λ = * , where * ∶ ℌ 1 →  is the usual Hilbert space adjoint. The operator Λ is called the Dirichletto-Neumann operator at the point = −1 associated with the boundary pair ( ,Γ 0 and then Λ( ) ∶= . The operator Λ( ) is closed in  and it has bounded inverse Λ( Next consider the restriction of * to the form domain of and equip it with the norm defined by ‖ ‖ 2 holds for all ∈ ℌ 1 0 , ∈ ℌ 1 ; this map is well defined by the formulas (2.35), (2.36). Finally, introduce the restriction * of * by holds for all ∈ dom * and ∈ ℌ 1 . In what follows the triple { , Γ 0 , Γ 1 } with the domain dom * = dom Γ 0 ∩ dom Γ 1 is called a boundary triple generated by the boundary pair ( ,Γ 0 ) . The next result characterizes the central properties of the boundary pair ( ,Γ 0 ) by means of the boundary triple { , Γ 0 , Γ 1 } . In particular, it shows that the notion of boundary pair in Definition 2.14 can be included in the framework of unitary boundary triples whose Weyl functions are Nevanlinna functions from the class  ().

APPLICATIONS TO LAPLACE OPERATORS
In this section the applicability of the abstract theory developed in the preceding sections is demonstrated for the analysis of some classes of differential operators. First we consider the most standard case of elliptic PDE by treating Laplacians in smooth bounded domains; in this case many of the abstract results take a rather explicit form.
With these preliminaries we are ready to describe complete analogs of the abstract results in Theorem 2.2 for Laplacians on smooth bounded domains. Let˜ * be a restriction of max to the domaiñ * = max ↾ dom˜ * , dom˜ * = 1∕2 Δ (Ω) =
(ii) More precisely, for every ∈ [3∕2, 2] the quasi boundary triples in (i) are, in fact, essentially unitary. The choice = 3∕2 in Theorem 3.1 is motivated by the following statement: for every ∈ [3∕2, 2] the closure of the graph of ( this closure is an -generalized boundary triple for max and, hence, it is unitary. (iii) It follows from (ii) that the Weyl function (⋅) of the quasi boundary triple Π for any ∈ (3∕2, 2) is not closed in 2 ( Ω), hence it is not a Nevanlinna function. However, the closure of (⋅) in 2 ( Ω) is just the Weyl function 3∕2 (⋅) = (⋅) of Π 3∕2 = Π in Theorem 3.1 (i). (i) General theory of, not necessarily local, boundary value problems for elliptic operators in bounded domains with smooth boundary was built in the pioneering works by Višik [65] and Grubb [39]. In terms of boundary triples Grubb's results were adapted and further developed in Malamud [57] (see also [1,14,21,36,40] for some further developments and applications). (ii) The description of the Kreȋn-von Neumann Laplacian (see Theorem 3.1 (ii)) in terms of boundary conditions for domains with smooth boundary is immediate by combining Krein's description of [54] with trace theory by Lions and Magenes (see [55]) and goes back to the works [65] and [60, Section 12.3] (see also [57]). For Lipschitz domains a similar description of the Kreȋn-von Neumann Laplacian in terms of extended trace operators was recently given in [10]; see also Section 3.3 below for another construction. (iii) Finally, it is mentioned that the abstract renormalization result in [26,Theorem 5.32], when specialized to the case of the -generalized boundary triple in Theorem 3.1 (ii), leads to an ordinary boundary triple for max ; for further discussion see [25,Cor. 7.7] and comments therein.

Mixed boundary value problem for Laplacian
Let Ω be a bounded open set in ℝ ( ≥ 2) with a smooth boundary Ω. Let Σ + be a compact smooth submanifold of Ω, Σ • + be the interior of Σ + and let Σ − ∶= Ω ⧵ Σ • + , so that Σ = Σ + ∪ Σ − . Let −Δ be the Zaremba Laplacian, i.e. the restriction of the maximal operator max to the set of functions, which satisfy Dirichlet boundary condition on Σ − and Neumann boundary condition on Σ + .
It is known (see for instance Grubb [41]), that the operator −Δ is associated with the nonnegative closed quadratic form hence it is selfadjoint in 0 (Ω). Clearly, its spectrum ( −Δ ) is discrete. Here we construct an -generalized boundary triple, associated with the Zaremba Laplacian. Let min and * be the minimal and pre-maximal operators, respectively, associated with −Δ, dom ( * ) = More precisely we have the following result. Proof.

Laplacians on Lipschitz domains
Here the smoothness properties on Ω are relaxed; it is assumed that Ω is a bounded Lipschitz domain. In this case the Dirichlet and Neumann traces are still continuous operators for all 1∕2 ≤ ≤ 3∕2 and, in addition, both are surjective when = 1∕2 and = 3∕2; see Gesztesy and Mitrea [36,Lemmas 3.1,3.2]. In this case the results, which are analogous to those in Section 3.1, will be derived directly from the abstract setting treated in Section 2.1.

Laplacian on rough domains
Let Ω be a bounded domain in ℝ ( ≥ 2) whose boundary Ω is equipped with a finite ( − 1)-dimensional Hausdorff measure , ( Ω) < ∞. To construct an analog for the boundary triple appearing in Theorem 3.1 (i) in nonsmooth domains Ω we make use of some results established in [23] and [4][5][6]. Following Arendt and ter Elst [4, Definition 3.1] we first recall the notion of a trace ∈ 2 ( ) for a class of functions ∈ 1 (Ω). Definition 3.10. A function ∈ 2 ( ) is said to be a trace of ∈ 1 (Ω), if there is a sequence Denote by 1 (Ω) the set of elements of 1 (Ω) for which there exists a trace. In general, the trace is not uniquely defined. It is possible that | Ω = 0 while its trace = | Ω in 2 ( ) is nontrivial; for an example see e.g. [4,Example 4.4]. Define the linear relation by Then can be considered as a mapping from 1 (Ω) to 2 ( ), which is linear but in general multivalued on the domain 1 (Ω) and it has dense range in 2 ( ); cf. [4]. If and are as in Definition 3.10 we shall write ∈ .
For associating an appropriate boundary triple in this setting, we impose the following additional assumption.
In [4, Definition 3.2] the (weak) normal derivative is defined implicitly via Green's (first) formula as follows: a function ∈ 1 (Ω) with Δ ∈ 2 (Ω) is said to have a weak normal derivative in 2 ( ) if there exists ∈ 2 ( ) such that , where Δ denotes the Laplacian understood in distributional sense. Since the functions , form a dense set in 2 ( ), the function ∈ 2 ( ) is uniquely determined by and the mapping → is denoted by : Assume that for some , ∈ 2 ( ), ∈ 1 (Ω), and ≤ 0 one has The operator Λ( ) which maps to is called the Dirichlet-to-Neumann map. A slight modification of the proof of [4,Theorem 3.3] shows, that Λ( ) is a nonnegative selfadjoint operator on 2 ( ) which is uniquely determined by the three properties listed in (3.26). Now consider the differential expression −Δ, where Δ = ∇ ⋅ ∇ is the (distributional) Laplacian operator in Ω. Recall (see [6,Example 3.1]) that for an open set Ω (without any regularity on the boundary) the Dirichlet Laplacian −Δ is defined as the selfadjoint operator associated with the closed (Dirichlet) form Similarly the Neumann Laplacian −Δ is defined as the selfadjoint operator associated with the closed form (see [6,Example 3.2]) . Then in view of (3.23) this identity can be extended to hold for all ∈ 1 (Ω). Thus, in particular, it holds for all ∶= ∈ dom Γ: Similarly, one gets from (3.25) with = ∈ dom Γ and = ∈ dom Γ: Taking conjugates in the last identity and subtracting the identity (3.30) from that leads to Green's (second) formula in [26, Equation (3.1)] for −Δ with , ∈ * = dom Γ. Thus, { 2 ( ), Γ } is an isometric boundary pair.

□
In this general setting, the multivalued part of Γ can be nontrivial, since the trace need not be uniquely determined. For unitary boundary pairs the multivalued part is described in [28,Lemma 4.1] and for isometric boundary pairs in [26,Lemma 3.6]. In the present setting a more explicit description of the multivalued part can be given with the aid of a result of Daners in [23]; see also [6] for an other proof of Daners result via capacity arguments. > 0, 0 can be considered to represent an irregular part of the boundary.
Remark 3.14. In this general setting we do not know if the operator 0 ∶= −Δ↾ ker Γ 0 coincides with the Dirichlet Laplacian −Δ . In other words, we do not know if the Neumann trace exists for every ∈ dom ( − Δ ) .
A criterion for a direct sum of ordinary boundary triples to form also an ordinary boundary triple can be formulated in terms of the corresponding Weyl functions (see [22,52,58]).  (iv) The direct sum Π = ⨁ ∞ =1 Π is an ordinary boundary triple for * if and only if in addition to (4.6) the following condition is fulfilled 5 The next result contains analogous characterization for -generalized boundary triples. Similarly, if the operators ( , 0 ) have a common gap ( − , + ), then Π forms an -generalized boundary triple for * if and only if 4 < ∞ where 4 is given by (4.6).
Proof. The condition (4.7) means that Im ( ) is bounded for some (equivalently for every) ∈ ℂ ± . By Theorem 1.3, this amounts to saying that Π is an -generalized boundary triple for * .
Similarly, in case of a common spectral gap ( − , + ) the condition (4.7) is equivalent to the condition 4 < ∞ in (4.6) as can be seen by the same argument that was used in [26,Remark 5.25]. □ The next result is immediate by combining Proposition 2.8 in (4.6) with Proposition 2.9.
In this case the transposed boundary triple Π ⊤ = { , Γ 1 , −Γ 0 } is -generalized. Similarly, the following conditions are equivalent: In this case the triple Π is a -generalized boundary triple.
Proof. By Theorem 4.1 (see [52,Theorem 3.2]) Π is a unitary boundary triple such that 0 = ker Γ 0 and 1 = ker Γ 1 are essentially selfadjoint. Now the first part of the statement follows easily from Proposition 2.9, while the second part is implied by Proposition 2.8. □  In quantum mechanics this operator in 1-D case appears in the form − ℏ , where ℏ = ℎ∕2 is the reduced Planck constant and whose eigenvalues are measuring the momentum of a particle.
(ii) Assume in addition, that * < ∞. Then the mapping ± can be extended to a bounded surjective mapping from . Moreover, the following estimate holds
Now we are ready to state and prove the main result of this subsection. } be the boundary triple for the operator * defined by (4.9).

(iii) The Weyl function (⋅) is domain invariant and its domain is given by
(iv) The domain of the form ( ) associated with the imaginary part Im ( ) is given by (4.14) (v) The triple Π is an -generalized boundary triple for * and 0 ≠ * 0 . Moreover, the imaginary part Im (⋅) of the Weyl function (⋅) takes values in () ⧵ (). Proof.
(ii) Assuming that * < ∞ it is shown in [58] that the triple Π = ⊕ ∈ℕ Π is an ordinary boundary triple for * if and only if * > 0. This result remains true also in the case * = ∞.
In accordance with Theorem 4.8 (iv), Hence the renormalization in [26,Theorem 5.32] is determined via the formulasΓ 0 = −1 Γ 0 ,Γ 1 = Γ 0 and the corresponding Weyl function is given by Since˜Im ( ) → 2 as → 0, we conclude that (the closure of) (⋅) is a bounded uniformly strict Nevanlinna function, (⋅) ∈ []. Thus, the renormalization procedure in this case leads to an ordinary boundary triple for * . In the case * < ∞ this renormalization procedure was firstly applied in [58] to construct the above mentioned ordinary boundary triple for * ; see Examples 3.2, 3.8 and Theorem 3.6 in [58]. ] .

Schrödinger operators with local point interactions
It is easily seen that a boundary triple Π = for H * can be chosen as ] . (4.16) The corresponding Weyl function is given by ] .
The converse statement is also true (see [52]): the condition * > 0 is necessary for the direct sum Π = ⊕ ∈ℕ Π to form a boundary triple for H max ∶= H * min . Such type of boundary triples have naturally arisen in investigation of spectral properties of the Hamiltonian H , associated in 2 ( ℝ + ) with a formal differential expression when treating H , as an extension of H min (see [51,52], and Remark 4.15 below).
Combining this fact with the first relation in (4.22) and noting that * = 0 and sin ( √ ) √ ∼ 1 as → 0, one concludes that the convergence of the series in (4.23) is equivalent to The proof is postponed after Lemma 4.12.
(iv) The proof is similar to that of the item (ii). First notice that Note that Im ( ) as a function of is bounded on the intervals [ , ∞), > 0. It follows from (4.17) that Hence the convergence of the series in (4.24) is equivalent to This proves the statement.
(vi) The proof is similar to that of the statement (ii). One should only use relations (4.25) instead of (4.22).
It remains to prove the assertion (iii) of Theorem 4.10. It is more involved and to this end we describe traces of functions ∈ 2,2 ( ℝ + ⧵ ) as well as traces of their first derivatives and prove an analog of Lemma 4.7.
Lemma 4.12. Let = { } ∞ =1 be as above and let 0 ≤ * ≤ * < ∞. Then the mapping )} ∈ℕ is well defined and bounded and its range ran Γ ′′ 0 is given by Proof. Denote temporarily the right-hand side of (4.26) by  ( . First we prove the inclusion .
. This inclusion implies ∈ 2,2 [ −1 , ] for each ∈ ℕ and, it is easy to check that . To complete the proof it remains to note that holds if and only if * > 0 (see [53]).
We are now ready to prove the assertion (iii) in Theorem 4.10, i.e., to prove relation (4.20).
Proof of item (iii) in Theorem 4.10. Let the righthand side of (4.20) be denoted temporarily by  0 ( Γ 0 ) . The inclusion ran To prove the reverse inclusion we choose any vector and consider the functions and = ⊕ ∞ 1 as defined in (4.31). As shown in Lemma 4.12 ∈ 2,2 ( ℝ + ⧵ ) and ′ satisfies the equalities (4.32). Besides, Note that the latter inclusion holds since * < ∞. Summing up we get Thus, ∈ dom Γ ′ 0 ∩ dom Γ ′ 1 = dom H * and this completes the proof. □ One gets from Lemma 4.12 a description for the ranges of the closures of Γ 0 and Γ 1 .
In other words, domain invariance property does not imply the property of a boundary triple to be -generalized (see also [26,Example 5.29]). Such Weyl functions cannot be written in the form (1.3) without a renormalization of the boundary triple as in [26,Theorem 5.32]. (ii) In the case * = 0 and * < ∞ an abstract regularization procedure from [52,58] has first been applied in [52] , is a Jacobi matrix. It is shown in [52] that certain spectral properties of H , strictly correlate with that of , .
Finally, it is mentioned that boundary triple models are motivated by and naturally appear in various physical problems as exactly solvable models that describe complicated physical phenomena; see, e.g., [3,14,35,53] is a -generalized boundary triple for * . Moreover,Π is an ordinary boundary triple if and only if * > 0. ] → ℂ 2 , ∈ ℕ, ∈ {0, 1} are given by is an ordinary boundary triple for * min .
The proof is similar to that of Theorem 4.10 and is omitted.

Dirac operators with local point interactions
Let be a differential expression acting on ℂ 2 -valued functions of a real variable. Here > 0 denotes the velocity of light and 1 , 2 , 3 stand for the Pauli matrices in ℂ 2 : .
Furthermore, let = { } ∞ 1 be a strictly increasing sequence of positive numbers satisfying lim →∞ = ∞, let , * , and * be defined by (4.8) and let be the minimal operator generated in 2 [ −1 , ] ⊗ ℂ 2 by the differential expression (4.36) Recall that is a symmetric operator with deficiency indices ± ( ) = 2 and its adjoint * is given by * = ↾ dom ( * ) , dom Next following [22] we recall the construction of a boundary triple for * and compute the corresponding Weyl function.
(i) The proof is immediate from Lemma 4.7.
The inclusion in (4.46) as well as the relation domΓ = domΓ 1 is implied by the first inclusion in (4.50). (iii) The Weyl function ofΠ is the direct sum˜(⋅) = ⨁ ∞ =1˜( ⋅), wherẽ ( ) ∶= This immediately leads to formula (4.47) for˜( ). Using (4.40), and the Taylor series expansions for sin( ) and cos( ) we easily derivẽ As a function of the imaginary part Im˜( ) is bounded on the intervals [ , ∞), > 0, and hence it follows from (4.51) that the convergence of the series in (4.52) is equivalent to where ( ) denotes the diagonal matrix function in the left-hand side of (4.51). Clearly, Im ( ) is bounded with bounded inverse for each ∈ ℂ ± and this yields the stated description of dom˜( ) . (v) By Theorem 4.1(iv), the tripleΠ being a direct sum of ordinary boundary triples, is an -generalized boundary triple. On the other hand, by (iii) the strict inclusion dom ( ) ⫋Γ 0 ( dom , * ) is equivalent to * = 0. Therefore, Theorem 1.3 applies and ensures that in the latter caseΠ is not an -generalized triple. Thus Π ⊤ is a -generalized boundary triple; cf. [33,Chapter 5]. □ Remark 4.20. Apart from statements (ii) and the formula forΓ 0 ( dom , * ) in statement (iii) the results in Proposition 4.19 remain valid for * = ∞. Indeed, statement (i) is still immediate from Proposition 4.7(i) which holds in this case, too. All the other statements can easily be extracted from the fact that the limit value of the Weyl function˜( ) as well as its inverse˜( ) −1 remain bounded when → ∞.

E N D N O T E
1 This notation is much better suited for applications than the notation in [26, (1.14)], where ( ) was used for the nonclosed form (1.4).