Scattering matrices and Dirichlet-to-Neumann maps
aa r X i v : . [ m a t h - ph ] J un SCATTERING MATRICES AND DIRICHLET-TO-NEUMANNMAPS
JUSSI BEHRNDT, MARK M. MALAMUD, AND HAGEN NEIDHARDT
Abstract.
A general representation formula for the scattering matrix of ascattering system consisting of two self-adjoint operators in terms of an ab-stract operator valued Titchmarsh-Weyl m -function is proved. This result isapplied to scattering problems for different self-adjoint realizations of Schr¨odin-ger operators on unbounded domains, Schr¨odinger operators with singular po-tentials supported on hypersurfaces, and orthogonal couplings of Schr¨odingeroperators. In these applications the scattering matrix is expressed in an ex-plicit form with the help of Dirichlet-to-Neumann maps. Introduction
Let A and B be self-adjoint operators in a Hilbert space H and assume that theresolvent difference( B − λ ) − − ( A − λ ) − , λ ∈ ρ ( A ) ∩ ρ ( B ) , (1.1)belongs to the ideal S ( H ) of trace class operators. It is well known that in thissituation the wave operators W ± ( A, B ) of the pair { A, B } exist and are complete,and the scattering operator S ( A, B ) = W + ( A, B ) ∗ W − ( A, B ) is unitarily equivalentto a multiplication operator induced by a family { S ( A, B ; λ ) } λ ∈ R of unitary oper-ators S ( A, B ; λ ) in the spectral representation of the absolutely continuous part of A . This family is called the scattering matrix of the scattering system { A, B } andis one of the most important quantities in the analysis of scattering processes; werefer the reader to the monographs [12, 59, 79, 81, 82] for more details.The main objective of this paper is to express the scattering matrix of { A, B } in terms of an abstract operator valued Titchmarsh-Weyl m -function, and to applythis result to scattering problems for Schr¨odinger operators. In order to explain ourmain abstract result Theorem 3.1 consider the closed symmetric operator S = A ∩ B and note that S has infinite defect numbers whenever the resolvent difference of A and B in (1.1) is infinite dimensional. The closure of the operator T = A b + B ,where b + denotes the sum of subspaces in H × H , coincides with S ∗ and clearly A and B are self-adjoint restrictions of T . This setting can be fitted in the framework of( B -)generalized boundary triples and their Weyl functions from [38] and allows tointroduce boundary maps Γ and Γ on dom( T ), which can be viewed as abstractanalogs of the Dirichlet and Neumann trace operators (see also [13, 14, 34, 35]).For λ ∈ C \ R one defines the Weyl function M via M ( λ )Γ f λ = Γ f λ , f λ ∈ ker( T − λ ) , see Section 2 for the details. In PDE applications M ( λ ) is usually the Dirichlet-to-Neumann map (or its inverse, the Neumann-to-Dirichlet map) acting in some boundary space. Roughly speaking our main abstract result states that the scat-tering matrix of { A, B } is of the form S ( A, B ; λ ) = I − i p Im M ( λ + i M ( λ + i − p Im M ( λ + i λ ∈ R . This representation is a highly nontrivial generalization of a similarresult from [19], where the special case that the resolvent difference in (1.1) is afinite rank operator was treated in the context of ordinary boundary triples andtheir Weyl functions from [37, 38], see also [2], [8, Chapter 4], [82, Chapter 3, § R and R inSection 4, and orthogonal couplings of Schr¨odinger operators, and Schr¨odinger op-erators with singular potentials supported on curves and hypersurfaces in R and R in Section 5. Let us first explain the situation for a scattering system consistingof a Neumann and a Robin realization; for more details and a slightly more generalsituation see Section 4.4. Denote the Dirichlet and Neumann trace operators by γ D and γ N , respectively, and consider the self-adjoint operators Af = − ∆ f + V f, dom( A ) = (cid:8) f ∈ H (Ω) : γ N f = 0 (cid:9) , and Bf = − ∆ f + V f, dom( B ) = (cid:8) f ∈ H (Ω) : αγ D f = γ N f (cid:9) , where α ∈ C ( ∂ Ω) is real, the potential V is real and bounded, and the domainΩ is the complement of a bounded set with a C ∞ -smooth boundary in R or R .In this situation it is known from [15, 58] that the resolvent difference of A and B satisfies the trace class condition (1.1). If N ( λ ), λ ∈ C \ R , denotes the Neumann-to-Dirichlet map, that is, N ( λ ) γ N f λ = γ D f λ , − ∆ f λ + V f λ = λf λ , we obtain in Theorem 4.7 that the scattering matrix of the scattering system { A, B } admits the form S ( A, B ; λ ) = I G λ + 2 i p Im N ( λ + i (cid:0) I − α N ( λ + i (cid:1) − α p Im N ( λ + i λ ∈ R . Here the space L ( R , dλ, G λ ), where G λ = ran(Im N ( λ + i λ ∈ R , forms a spectral representation of the absolutely continuous part ofthe Neumann operator A N and the limits Im N ( λ + i
0) and ( I − α N ( λ + i − have to be interpreted in suitable operator topologies; cf. Theorem 4.7 for details.A similar result is proved in Theorem 4.3 for the pair consisting of the Dirichletrealization of − ∆ + V and the Robin operator B in L ( R ); here the trace classproperty (1.1) for n = 2 is due to Birman [24]. For some recent work on relatedspectral problems for Schr¨odinger operators we refer the reader to [9, 22, 30, 46, 47,48, 49, 50, 64, 67, 74, 77] and for more general partial elliptic differential operatorsto [1, 13, 14, 17, 18, 21, 29, 55, 56, 57, 58, 63, 65, 66, 75, 76]. CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 3
Our second set of examples in Section 5 is a bit more involved. Here scatteringsystems consisting of the free Schr¨odinger operator Af = − ∆ f + V f, dom( A ) = H ( R n ) , (1.2)and orthogonal couplings of Schr¨odinger operators with Dirichlet and Neumannboundary conditions, or Schr¨odinger operators with singular δ -potentials of strength α ∈ L ∞ ( C ) supported on hypersurfaces C which split R or R into a boundedsmooth domain Ω + and a smooth exterior domain Ω − are studied. The latteroperator is of the form Bf = − ∆ f + V f, dom( B ) = (cid:26) f = (cid:18) f + f − (cid:19) ∈ H / ( R n \ C ) : γ + D f + = γ − D f − ,αγ ± D f ± = γ + N f + + γ − N f − (cid:27) ; (1.3)here H / ( R n \ C ) is a subspace of H / (Ω + ) × H / (Ω − ) and γ ± D and γ ± N denotethe Dirichlet and Neumann trace operators on the interior and exterior domain;cf. Section 5.4 for the details. Schr¨odinger operators with δ -potentials play animportant role in various physically relevant problems and have therefore attracteda lot of attention. We refer the interested reader to the review paper [39], to e.g.[7, 10, 16, 27, 40, 41, 42, 43] and the monographs [6, 8] for more details and furtherreferences. We shall briefly discuss the scattering matrix for the pair of operatorsin (1.2)–(1.3); for the pairs consisting of A in (1.2) and the orthogonal sum of theDirichlet or the Neumann realizations of − ∆ + V on Ω + and Ω − see Theorem 5.1and Theorem 5.4, respectively. It follows from [16] that the above choice of A and B satisfies the trace class condition (1.1) in dimensions n = 2 and n = 3 and weshow in this situation in Theorem 5.6 that the scattering matrix is given by S ( A, B ; λ ) = I G λ + 2 i p Im E ( λ + i (cid:0) I − α E ( λ + i (cid:1) − α p Im E ( λ + i , where the function E is defined as E ( λ ) = (cid:0) D + ( λ ) − + D − ( λ ) − (cid:1) − , λ ∈ C \ R , and D ± ( λ ) denote the Dirichlet-to-Neumann maps corresponding to − ∆ + V onthe domains Ω ± . In this context we also refer the reader to related work by B.S.Pavlov and coauthors in [11, 69, 72], where scattering problems for certain couplingsof Schr¨odinger operators were considered. Notation.
Throughout the paper H and G denote separable Hilbert spaces withscalar product ( · , · ). The linear space of bounded linear operators defined from H to G is denoted by B ( H , G ). For brevity we write B ( H ) instead of B ( H , H ). Theideal of compact operators is denoted by S ∞ ( H , G ) and S ∞ ( H ). For p > S p ( H , G ) and S p ( H ); they consist ofall compact operators T with p -summable singular values s j ( T ) (i.e. eigenvalues of( T ∗ T ) / ). We shall also work with the operator ideals S p ( H , G ) = (cid:8) T ∈ S ∞ ( H , G ) | s j ( T ) = O ( j − /p ) as j → ∞ (cid:9) , p > , and we recall that S p ( H , G ) · S q ( H , G ) = S r ( H , G ) , where 1 p + 1 q = 1 r . (1.4)The resolvent set and the spectrum of a linear operator A is denoted by ρ ( A )and σ ( A ), respectively. The domain, kernel and range of a linear operator A are J. BEHRNDT, M. M. MALAMUD, AND H. NEIDHARDT denoted by dom( A ), ker( A ), and ran( A ), respectively. By B ( R ) we denote theBorel sets of R . The Lebesgue measure on B ( R ) is denoted by dλ .A holomorphic function M ( · ) : C + −→ B ( H ) is a Nevanlinna (or Herglotz or R -function) if its imaginary part Im( M ( z )) := i ( M ( z ) − M ( z ) ∗ ), z ∈ C + , is a non-negative operator. Nevanlinna functions are extended to C − by M ( z ) := M (¯ z ) ∗ , z ∈ C − . The class of B ( H )-valued Nevanlinna functions is denoted by R [ H ]. ANevanlinna function satisfying ker(Im( M ( z )) = { } (0 ∈ ρ (Im( M ( z ))) for some,and hence for all, z ∈ C + , is said to be strict (uniformly strict, respectively). Thesesubclasses are denoted by R s [ H ] and R u [ H ], respectively.2. Self-adjoint extensions of symmetric operators and abstractTitchmarsh-Weyl m -functions In the preparatory Section 2.1 we recall the notion of boundary triples and theirWeyl functions from extension theory of symmetric operators, and we introduce theconcept of S p -regular Weyl functions in Section 2.2. This notion is important anduseful for our purposes since it is directly related (and in some situations equivalent)to the S p -property of the resolvent difference of certain self-adjoint extensions.2.1. B -generalized boundary triples and their Weyl functions. In this sub-section we review the notion of generalized (or B -generalized) and ordinary bound-ary triples from extension theory of symmetric operators, and we introduce a newconcept, the so-called double B -generalized boundary triples in Definition 2.1 be-low. We refer the reader to [28, 31, 34, 37, 38, 51, 80] for more details on ordinaryand B -generalized boundary triples, see also [13, 14, 32] for related notions.In the following S denotes a densely defined, closed, symmetric operator in aseparable Hilbert space H . Definition 2.1 ([38]) . A triple
Π = {H , Γ , Γ } is called a B -generalized boundarytriple for S ∗ if H is a Hilbert space and for some operator T in H such that T = S ∗ ,the linear mappings Γ , Γ : dom( T ) −→ H satisfy the abstract Green’s identity ( T f, g ) − ( f, T g ) = (Γ f, Γ g ) − (Γ f, Γ g ) , f, g ∈ dom( T ) , (2.1) the operator A := T ↾ ker(Γ ) is self-adjoint in H , and ran(Γ ) = H holds.If, in addition, the operator A := T ↾ ker(Γ ) is self-adjoint in H and ran(Γ ) = H , then the triple Π = {H , Γ , Γ } is called a double B -generalized boundary triple for S ∗ . We note that a B -generalized boundary triple for S ∗ exists if and only if S admits self-adjoint extensions in H , that is, the deficiency indices of S coincide.Furthermore, if Π = {H , Γ , Γ } is a B -generalized boundary triple for S ∗ thendom( S ) = ker(Γ ) ∩ ker(Γ )holds, the mappings Γ , Γ : dom( T ) −→ H are closable when viewed as linearoperators from dom S ∗ equipped with the graph norm to H , and ran(Γ ) turns outto be dense in H ; cf. [38, Section 6]The notion of double B -generalized boundary triples is inspired by the fact thatthe mappings in the so-called transposed triple Π ⊤ := {H , Γ , − Γ } satisfy theabstract Green’s identity but since in general neither A = T ↾ ker(Γ ) is self-adjointnor ran(Γ ) = H holds the transposed triple Π ⊤ is not a B -generalized boundarytriple in general. In fact, a B -generalized boundary triple Π = {H , Γ , Γ } for S ∗ is CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 5 a double B -generalized boundary triple for S ∗ if and only if the transposed tripleΠ ⊤ = {H , Γ , − Γ } is also a B -generalized boundary triple for S ∗ .In some of the proofs of the results in Section 2.2 we shall also make use of thenotion of ordinary boundary triples, which we recall here for the convenience of thereader. Definition 2.2.
A triple
Π = {H , Γ , Γ } is called an ordinary boundary triple for S ∗ if H is a Hilbert space, the linear mappings Γ , Γ : dom( S ∗ ) −→ H satisfy theabstract Green’s identity ( S ∗ f, g ) − ( f, S ∗ g ) = (Γ f, Γ g ) − (Γ f, Γ g ) , f, g ∈ dom( S ∗ ) , (2.2) and the mapping Γ = (Γ , Γ ) ⊤ : dom( S ∗ ) → H × H is surjective. Observe that any ordinary boundary triple is automatically a double B -genera-lized boundary triple; the converse is not true in general. Ordinary boundary triplesare an efficient tool in extension theory of symmetric operators. In particular, ifΠ = {H , Γ , Γ } is an ordinary boundary triple for S ∗ , then all closed properextensions e S ⊂ S ∗ of S in H can be parametrized by means of the set of closedlinear relations in H via e S Θ := (cid:8) { Γ f, Γ f } : f ∈ dom( e S ) (cid:9) ⊂ H × H (2.3)We write e S = S Θ . If Θ is an operator then (2.3) takes the form S Θ = S ∗ ↾ ker(Γ − ΘΓ )One verifies ( S Θ ) ∗ = S Θ ∗ and hence the self-adjoint extensions of S in H correspondto the self-adjoint relations Θ in H . We shall use that Θ in (2.3) is an operator(and not a multivalued linear relation) if and only if the extension S Θ and A = S ∗ ↾ ker(Γ ) are disjoint, that is, A ∩ S Θ = S .Next we recall the notions and some important properties of γ -fields and Weylfunctions. For an ordinary boundary triple they go back to [36, 37], for B -genera-lized boundary triples we refer the reader to [38]. In the following let {H , Γ , Γ } bea B -generalized boundary triple for S ∗ ; the special case of an ordinary boundarytriple is then covered as well. Observe first that for each z ∈ ρ ( A ), A = T ↾ ker(Γ ), the following direct sum decomposition holdsdom( T ) = dom( A ) ˙+ ker( T − z ) = ker(Γ ) ˙+ ker( T − z ) . (2.4)Hence the restriction of the mapping Γ to ker( T − z ) is injective. Definition 2.3 ([38]) . Let
Π = {H , Γ , Γ } be a B -generalized boundary triple.The γ -field γ ( · ) and the Weyl function M ( · ) corresponding to Π are defined by γ ( z ) := (cid:0) Γ ↾ ker( T − z ) (cid:1) − and M ( z ) := Γ γ ( z ) , z ∈ ρ ( A ) , respectively. It follows from (2.4) that for z ∈ ρ ( A ) the values γ ( z ) of the γ -field and thevalues M ( z ) of the Weyl function are both well defined linear operators on ran(Γ ) = H . Moreover, γ ( z ) ∈ B ( H , H ) maps onto ker( T − z ) ⊂ ker( S ∗ − z ) ⊂ H and for all z, ξ ∈ ρ ( A ) the relations γ ( z ) = (cid:0) I + ( z − ξ )( A − z ) − (cid:1) γ ( ξ ) = ( A − ξ )( A − z ) − γ ( ξ ) (2.5)and γ ( z ) ∗ = Γ ( A − ¯ z ) − ∈ B ( H , H ) (2.6) J. BEHRNDT, M. M. MALAMUD, AND H. NEIDHARDT hold. In particular, ran( γ ( z ) ∗ ) = ran(Γ ↾ dom( A )) does not depend on the point z ∈ ρ ( A ) and (cid:0) ran γ ( z ) ∗ (cid:1) ⊥ = ker γ ( z ) = { } shows that ran( γ ( z ) ∗ ) is dense in H for all z ∈ ρ ( A ). Furthermore, it follows from(2.5) that γ ( · ) is holomorphic on ρ ( A ).The values of the Weyl function M ( · ) are operators in B ( H ) and M ( z ) maps H into the dense subspace ran(Γ ) ⊂ H . The Weyl function and the γ -field arerelated by the identity M ( z ) − M ( ξ ) ∗ = ( z − ¯ ξ ) γ ( ξ ) ∗ γ ( z ) , z, ξ ∈ ρ ( A ) , (2.7)and, in particular, M (¯ z ) = M ( z ) ∗ for all z ∈ ρ ( A ). It follows from (2.5) and (2.7)that M ( · ) is holomorphic on ρ ( A ). Setting ξ = z in (2.7) one getsIm M ( z ) = 12 i ( M ( z ) − M ( z ) ∗ ) = (Im z ) γ ( z ) ∗ γ ( z ) (2.8)and hence Im M ( z ) ≥ z ∈ C + . This identity also yieldsker(Im M ( z )) = ker( γ ( z )) = { } , z ∈ C ± , and together with the holomorphy of M ( · ) on ρ ( A ) we conclude that M ( · ) is a so-called strict Nevanlinna function with values in B ( H ); we shall denote this by M ( · ) ∈ R s [ H ]. If Π is a double B -generalized boundary triple then the Weyl functioncorresponding to the transposed B -generalized boundary triple Π ⊤ = {H , Γ , − Γ } is given by − M ( · ) − and also belongs to the class R s [ H ], in particular, for z ∈ ρ ( A ) ∩ ρ ( A ) the values M ( z ) of the Weyl function of a double B -generalizedboundary triple are bounded and boundedly invertible operators.If Π is an ordinary boundary triple then the operators γ ( z ) are boundedly in-vertible when viewed as operators from H onto ker( S ∗ − z ). In this case it followsfrom (2.8) that Im M ( z ) is a uniformly positive operator for z ∈ C + , and hence theWeyl function corresponding to an ordinary boundary triple belongs to the class R u [ H ] of the so-called uniformly strict Nevanlinna functions with values in B ( H );cf. [34].2.2. Resolvent comparability and S p -regular Weyl functions. Let Π = {H , Γ , Γ } be a B -generalized boundary triple for S ∗ with the corresponding Weylfunction M ( · ), and let A = S ∗ ↾ ker(Γ ) and A = S ∗ ↾ ker(Γ ). It is importantto characterize the property of the resolvent comparability of the operators A and A in terms of the Weyl function M ( · ). To this end we introduce the notion of S p -regular Nevanlinna functions in the next definition. Definition 2.4.
A Nevanlinna function M ( · ) ∈ R [ H ] is called S p -regular for some p ∈ (0 , ∞ ] if it admits a representation M ( z ) = C + K ( z ) , K ( · ) : C + −→ S p ( H ) , z ∈ C + , (2.9) where C ∈ B ( H ) is a self-adjoint operator such that ∈ ρ ( C ) and K ( · ) is a strictNevanlinna function with values in B ( H ) , that is, K ( · ) ∈ R s [ H ] . The class of S p -regular Nevanlinna functions is denoted by R reg S p [ H ] . In other words, a Nevanlinna function is S p -regular if it differs from a strictNevanlinna function with values in S p by a bounded and boundedly invertibleself-adjoint constant. Lemma 2.5. If M ( · ) ∈ R reg S p [ H ] for some p ∈ (0 , ∞ ] , then − M ( · ) − ∈ R reg S p [ H ] . CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 7
Proof.
Since M ( · ) ∈ R reg S p [ H ] for some p ∈ (0 , ∞ ], there exists a boundedly invertibleself-adjoint operator C and a strict Nevanlinna function K ( · ) ∈ R s [ H ] such that M ( z ) = C + K ( z ) , z ∈ C + . (2.10)Observe first that ker( M ( z )) = { } holds for all z ∈ C + . In fact, M ( z ) ϕ = 0 yields(( C + Re K ( z )) ϕ, ϕ ) = 0 and (Im K ( z ) ϕ, ϕ ) = 0, and as K ( · ) is strict we conclude ϕ = 0 from the latter. Furthermore, as 0 ∈ ρ ( C ) and K ( z ) ∈ S p ( H ) it followsfrom the Fredholm alternative (see, e.g. [78, Corollary to Theorem VI.14]) that0 ∈ ρ ( M ( z )) for all z ∈ C + . It is clear that − M ( z ) − = D + L ( z ) , z ∈ C + , (2.11)holds with L ( z ) := C − − M ( z ) − , z ∈ C + and the boundedly invertible self-adjointoperator D := − C − . Since L ( z ) = C − − M ( z ) − = C − K ( z ) M ( z ) − , z ∈ C + , and K ( z ) ∈ S p ( H ) we conclude L ( z ) ∈ S p ( H ), z ∈ C + . Moreover, as C − is abounded self-adjoint operator one getsIm L ( z ) = Im (cid:0) − M ( z ) − (cid:1) = ( M ( z ) ∗ ) − (cid:0) Im K ( z ) (cid:1) M ( z ) − , z ∈ C + , where in the last equality we have used (2.10). As K ( · ) ∈ R s [ H ] by assumption wehave ker(Im K ( z )) = { } and this yields ker(Im L ( z )) = { } for all z ∈ C + . Wehave shown that L ( · ) : C + −→ S p ( H ) is a strict Nevanlinna function, L ( · ) ∈ R s [ H ],and hence it follows from (2.11) that − M − ( · ) ∈ R reg S p [ H ]. (cid:3) The assertions in the next lemma on the boundary values of S -regular Nevan-linna functions follow from well-known results due to Birman and `Entina [25], deBranges [26], and Naboko [70]; cf. [44, Theorem 2.2]. Lemma 2.6.
Let M ( · ) be an S -regular Nevanlinna function, M ( · ) ∈ R reg S [ H ] .Then the following assertions hold. (i) M ( λ + i
0) = lim ε → +0 M ( λ + iε ) exists for a.e. λ ∈ R in the norm of B ( H ) ; (ii) M ( λ + i is boundedly invertible in H for a.e. λ ∈ R ; (iii) M ( λ + iε ) − M ( λ + i ∈ S p ( H ) for p ∈ (1 , ∞ ] , ε > and a.e. λ ∈ R , and lim ε → +0 k M ( λ + iε ) − M ( λ + i k S p ( H ) = 0;(iv) Im M ( λ + i
0) = lim ε → +0 Im M ( λ + iε ) exists for a.e. λ ∈ R in the S -norm.Proof. By assumption there exists a Nevanlinna function K ( · ) with values in S ( H )such that M ( z ) = C + K ( z ), z ∈ C + , holds with some bounded and boundedlyinvertible self-adjoint operator C . It follows from [25, 26, 70] (see, e.g. [44, Theorem2.2]) that the limit K ( λ + i
0) exists for a.e. λ ∈ R in the S p -norm for all p > K ( λ + i
0) exists for a.e. λ ∈ R in the S -norm. This yieldsassertions (i), (iii), and (iv).In order to prove (ii) we recall that − M ( · ) − is S -regular by Lemma 2.5 andhence the boundary values M ( λ + i − exist for a.e. λ ∈ R in the operator norm.Hence (ii) follows from the identity M ( λ + iε ) M ( λ + iε ) − = M ( λ + iε ) − M ( λ + iε ) = I H , λ ∈ R , after passing to the limit ε → +0 in the operator norm. (cid:3) J. BEHRNDT, M. M. MALAMUD, AND H. NEIDHARDT
In the next lemma we investigate B -generalized boundary triples with S p -regularWeyl functions. In particular, it turns out that the symmetric extension A = T ↾ ker(Γ ) is self-adjoint and a Krein type resolvent formula is obtained; cf. [14, 17,37, 38]. Proposition 2.7.
Let
Π = {H , Γ , Γ } be a B -generalized boundary triple for S ∗ such that the corresponding Weyl function M ( · ) is S p -regular for some p ∈ (0 , ∞ ] .Then the following assertions hold. (i) Π is a double B -generalized boundary triple for S ∗ ; (ii) The Weyl function corresponding to the transposed B -generalized boundarytriple Π ⊤ = {H , Γ , − Γ } is S p -regular; (iii) The operators A and A are S p -resolvent comparable and ( A − z ) − − ( A − z ) − = − γ ( z ) M ( z ) − γ (¯ z ) ∗ ∈ S p ( H ) (2.12) holds for all z ∈ ρ ( A ) ∩ ρ ( A ) .Proof. (i) Since the Weyl function M ( · ) is S p -regular by assumption, Lemma 2.5implies, in particular, that M ( z ) − ∈ B ( H ) for all z ∈ C \ R . This yields ran(Γ ) =ran( M ( z )) = H . Next we check that A = T ↾ ker(Γ ) is self-adjoint in H . Firstof all it follows from the abstract Green’s identity (2.1) that A is symmetric. Let z ∈ C \ R , fix f ∈ H and consider h := ( A − z ) − f − γ ( z ) M ( z ) − γ (¯ z ) ∗ f. From Definition 2.3 and (2.6) we obtainΓ h = Γ ( A − z ) − f − Γ γ ( z ) M ( z ) − γ (¯ z ) ∗ f = 0and hence h ∈ dom( A ). Since ran γ ( z ) ⊂ ker( T − z )) one gets( A − z ) h = ( T − z ) (cid:0) ( A − z ) − f − γ ( z ) M ( z ) − γ (¯ z ) ∗ f (cid:1) = f and we conclude the Krein type resolvent formula (2.12) in (iii) and ran( A − z ) = H for z ∈ C \ R . Hence the symmetric operator A is self-adjoint in H and it followsthat Π is a double B -generalized boundary triple for S ∗ .(ii) The Weyl function corresponding to the transposed B -generalized boundarytriple Π ⊤ = {H , Γ , − Γ } is given by M ⊤ ( z ) = − M ( z ) − , z ∈ ρ ( A ) ∩ ρ ( A ) , (2.13)which is S p -regular by Lemma 2.5.(iii) Since M ( · ) is S p -regular it follows that Im M ( z ) ∈ S p ( H ) for z ∈ C \ R and hence γ ( z ) ∗ γ ( z ) ∈ S p ( H ) by (2.8). This implies γ ( z ) ∈ S p ( H , H ) and γ ( z ) ∗ ∈ S p ( H , H ) for z ∈ C \ R , and the resolvent formula in (2.12) together with0 ∈ ρ ( M ( z )), z ∈ C \ R , yields the S p -property of the resolvent difference in (2.12)for z ∈ C \ R , and hence for all z ∈ ρ ( A ) ∩ ρ ( A ). (cid:3) Proposition 2.7 (iii) admits the following useful improvement.
Corollary 2.8.
Let
Π = {H , Γ , Γ } be a B -generalized boundary triple for S ∗ such that the corresponding Weyl function M ( · ) is S ∞ -regular and assume that Im M ( z ) ∈ S p ( H ) for some p ∈ (0 , ∞ ) and z ∈ C + . Then ( A − z ) − − ( A − z ) − ∈ S p ( H ) , z ∈ ρ ( A ) ∩ ρ ( A ) . (2.14) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 9
Proof.
The assumption Im M ( z ) ∈ S p ( H ) for some p ∈ (0 , ∞ ) and z ∈ C + togetherwith (2.8) yields γ ( z ) ∗ γ ( z ) ∈ S p ( H ), and hence γ ( z ) ∈ S p ( H , H ). The Krein typeformula in (2.12) implies (2.14) for z ∈ C + , and hence also for all z ∈ ρ ( A ) ∩ ρ ( A ). (cid:3) Next we show that the p -resolvent comparability condition (2.12) guaranteesthe existence of a B -generalized boundary triple such that the corresponding Weylfunction is S p -regular. Proposition 2.9.
Let A and B be self-adjoint operators in H and assume that theclosed symmetric operator S = A ∩ B is densely defined. Then dom( A ) + dom( B ) is dense in dom( S ∗ ) with respect to the graph norm and the following assertionshold. (i) There is a B -generalized boundary triple Π = {H , Γ , Γ } for S ∗ such that A = T ↾ ker(Γ ) = A and B = T ↾ ker(Γ ) = A . (2.15)(ii) If for some z ∈ C \ R and some p ∈ (0 , ∞ ] the condition ( B − z ) − − ( A − z ) − ∈ S p ( H ) (2.16) is satisfied, then there exists a double B -generalized boundary triple Π = {H , Γ , Γ } such that (2.15) holds and the corresponding Weyl function M ( · ) is S p -regular.Proof. In order to see that dom( A ) + dom( B ) is dense in dom( S ∗ ) with respect tothe graph norm assume that h ∈ dom( S ∗ ) is such that( f A + f B , h ) + (cid:0) S ∗ ( f A + f B ) , S ∗ h (cid:1) = 0 for all f A ∈ dom( A ) , f B ∈ dom( B ) . Then ( Af A , S ∗ h ) = ( f A , − h ) and ( Bf B , S ∗ h ) = ( f B , − h ) for all f A ∈ dom( A ) and f B ∈ dom( B ) yield S ∗ h ∈ dom( A ) ∩ dom( B ) = dom( S ) and ( I + SS ∗ ) h = 0. Sincethe operator I + SS ∗ is uniformly positive one gets h = 0, that is, dom( A )+dom( B )is dense in dom( S ∗ ) with respect to the graph norm.(i) Observe first that S = A ∩ B is a densely defined, closed, symmetric operatorwith equal deficiency indices. Hence there exists an ordinary boundary triple Π ′ = {H , Γ ′ , Γ ′ } for S ∗ such that B = S ∗ ↾ ker(Γ ′ ); cf. [38, 36]. Furthermore, as A and B are disjoint self-adjoint extensions of S there exists a self-adjoint operatorΘ = Θ ∗ ∈ C ( H ) such that A = S ∗ ↾ dom( A ) , dom( A ) = ker(Γ ′ − ΘΓ ′ ) , see, e.g. [38, Proposition 1.4]. We consider the mappingsΓ := Γ ′ − ΘΓ ′ and Γ := − Γ ′ defined on dom(Γ ) = dom(Γ ) := dom( A ) + dom( B ) , and set T := S ∗ ↾ dom( T ) , dom( T ) := dom( A ) + dom( B ) . We claim that Π = {H , Γ , Γ } is a B -generalized boundary triple for S ∗ such that(2.15) holds. Note first that A = T ↾ ker(Γ ) = A , B = T ↾ ker(Γ ) = A , and that A and B are disjoint self-adjoint extensions of S by construction. Therefore theargument in the beginning of the proof implies that dom( T ) = dom( A ) + dom( B ) is dense in dom( S ∗ ) equipped with the graph norm and hence T = S ∗ . Moreover, sinceΘ = Θ ∗ and the abstract Green’s identity (2.2) holds for the ordinary boundarytriple Π ′ we obtain for f, g ∈ dom( T )(Γ f, Γ g ) − (Γ f, Γ g ) = (cid:0) − Γ ′ f, (Γ ′ − ΘΓ ′ ) g (cid:1) − (cid:0) (Γ ′ − ΘΓ ′ ) f, − Γ ′ g (cid:1) = (Γ ′ f, Γ ′ g ) − (Γ ′ f, Γ ′ g ) = ( T f, g ) − ( f, T g ) , that is, the abstract Green’s identity (2.1) holds. In order to verify ran(Γ ) = H fix h ∈ H . Since Π ′ is an ordinary boundary triple there exists f ∈ dom( B ) = ker(Γ ′ )such that Γ ′ f = h . We then obtainΓ f = (Γ ′ − ΘΓ ′ ) f = Γ ′ f = h, and hence ran(Γ ) = H . Summing up, we have shown that Π is a B -generalizedboundary triple such that (2.15) holds.(ii) Now we choose an ordinary boundary triple Π ′′ = {H , Γ ′′ , Γ ′′ } for S ∗ suchthat A = S ∗ ↾ ker(Γ ′′ ). Since A and B are disjoint extensions of S there exists anoperator Θ = Θ ∗ ∈ C ( H ) such that B = S ∗ ↾ dom( B ) , dom( B ) = ker(Γ ′′ − ΘΓ ′′ ) . (2.17)It follows from [37, Theorem 2] that the condition (2.16) is equivalent to the con-dition (Θ − ξ ) − ∈ S p ( H ) for all ξ ∈ ρ (Θ). In particular, ρ (Θ) ∩ R = ∅ , and in thefollowing we assume without loss of generality that 0 ∈ ρ (Θ). Denote the spectralfunction of the self-adjoint operator Θ by E Θ ( · ), let sgn(Θ) = R R sgn( t ) dE Θ ( t ) andrecall the polar decompositionΘ = | Θ | / sgn(Θ) | Θ | / = sgn(Θ) | Θ | = | Θ | sgn(Θ) . As Θ − ∈ S p ( H ) we have | Θ | − / ∈ S p ( H ) and ker( | Θ | − / ) = { } . We considerthe mappings Γ := | Θ | / Γ ′′ and Γ := | Θ | − / (Γ ′′ − ΘΓ ′′ ) (2.18)defined ondom(Γ ) = dom(Γ ) := (cid:8) f ∈ dom( S ∗ ) : Γ ′′ f ∈ dom( | Θ | / ) (cid:9) . (2.19)We set T := S ∗ ↾ dom( T ) , dom( T ) := dom(Γ ) = dom(Γ ) , and we claim that Π = {H , Γ , Γ } is a double B -generalized boundary triple for S ∗ . First of all we have for f, g ∈ dom( T )(Γ f, Γ g ) − (Γ f, Γ g )= (cid:0) | Θ | − / (Γ ′′ − ΘΓ ′′ ) f, | Θ | / Γ ′′ g (cid:1) − (cid:0) | Θ | / Γ ′′ f, | Θ | − / (Γ ′′ − ΘΓ ′′ ) g (cid:1) = (cid:0) (Γ ′′ − ΘΓ ′′ ) f, Γ ′′ g (cid:1) − (cid:0) Γ ′′ f, (Γ ′′ − ΘΓ ′′ ) g (cid:1) = (Γ ′′ f, Γ ′′ g ) − (Γ ′′ f, Γ ′′ g )and since Π ′′ is an ordinary boundary triple the abstract Green’s identity (2.1)follows. The condition ran(Γ ) = H is satisfied since 0 ∈ ρ (Θ), and thus also0 ∈ ρ ( | Θ | / ). It is also clear from the definition of Γ in (2.18)-(2.19) thatker(Γ ) = ker(Γ ′′ ) = dom( A ) . (2.20)Next it will be shown that ker(Γ ) = dom( B ) (2.21) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 11 holds. In fact, the inclusion ker(Γ ) ⊂ dom( B ) in (2.21) follows from the definitionof Γ in (2.18)-(2.19) and ker( | Θ | − / ) = { } . For the remaining inclusion let f ∈ dom( B ). Then Γ ′′ f = ΘΓ ′′ f by (2.17) and, in particular,Γ ′′ f ∈ dom(Θ) ⊂ dom( | Θ | / ) . Hence dom( B ) ⊂ dom( T ) and Γ f = 0 is clear, that is, dom( B ) ⊂ ker(Γ ) andthus (2.21) is shown. Combining (2.20) with (2.21) yields (2.15). Moreover, wehave T = S ∗ sincedom( A ) + dom( B ) = ker(Γ ) + ker(Γ ) ⊂ dom( T )and dom( A ) + dom( B ) is dense in dom( S ∗ ) equipped with the graph norm (as A and B are disjoint self-adjoint extensions of S ). Summing up, we have shown thatΠ = {H , Γ , Γ } is a B -generalized boundary triple for S ∗ such that (2.15) holds.It remains to verify that the Weyl function corresponding to Π is S p -regular;Proposition 2.7 (i) then implies that Π is a double B -generalized boundary triple.For this denote the Weyl function corresponding to the ordinary boundary tripleΠ ′′ by M ′′ ( · ) and recall that M ′′ ( z )Γ ′′ f z = Γ ′′ f z for f z ∈ ker( S ∗ − z ) and z ∈ ρ ( A ).We claim that the Weyl function corresponding to Π is given by M ( z ) = | Θ | − / M ′′ ( z ) | Θ | − / − sgn(Θ) , z ∈ ρ ( A ) . (2.22)In fact, for f z ∈ ker( T − z ) we compute (cid:0) | Θ | − / M ′′ ( z ) | Θ | − / − sgn(Θ) (cid:1) Γ f z = | Θ | − / M ′′ ( z )Γ ′′ f z − sgn(Θ) | Θ | / Γ ′′ f z = | Θ | − / (cid:0) Γ ′′ f z − | Θ | / sgn(Θ) | Θ | / Γ ′′ f z (cid:1) = | Θ | − / (cid:0) Γ ′′ f z − ΘΓ ′′ f z (cid:1) = Γ f z and hence (2.22) follows by Definition 2.3. Let K ( z ) := | Θ | − / M ′′ ( z ) | Θ | − / , z ∈ C + and let C := − sgn(Θ). Note that C is a boundedly invertible self-adjointoperator and that | Θ | − / ∈ S p ( H ) and M ′′ ( z ) ∈ B ( H ) yield K ( z ) ∈ S p ( H ), z ∈ C + . Moreover, as M ′′ ( · ) ∈ R u [ H ] it follows that K ( · ) ∈ R s [ H ], and hence theWeyl function M ( · ) is S p -regular. (cid:3) In applications to scattering problems it is important to know whether the resol-vent p -comparability condition (2.12), (2.16) yields the S p -regularity of the Weylfunction. Apparently a converse statement to Proposition 2.7 is false for arbitrarydouble B -generalized boundary triples, while Proposition 2.9 ensures the existenceof such a double B -generalized boundary triple. However in the following propo-sition we present an affirmative answer to this question under certain additionalexplicit assumptions. Proposition 2.10.
Let A and B be self-adjoint operators in H such that R B,A ( z ) := ( B − z ) − − ( A − z ) − ∈ S p ( H ) (2.23) holds for some z ∈ C \ R and some p ∈ (0 , ∞ ] , and assume that the closed symmetricoperator S = A ∩ B is densely defined. Assume, in addition, that there exists λ ∈ ρ ( A ) ∩ ρ ( B ) ∩ R such that ± R B,A ( λ ) > . (2.24) If Π = {H , Γ , Γ } is a double B -generalized boundary triple for S ∗ such that con-dition (2.15) holds then the corresponding Weyl function M ( · ) is S p -regular. Proof.
Since Π is a double B -generalized boundary triple the values of the Weylfunction M ( · ) and the function − M ( · ) − are in B ( H ). Moreover, the assumption λ ∈ ρ ( A ) ∩ ρ ( B ) ∩ R ensures that − M ( λ ) − ∈ B ( H ) is a self-adjoint operator andwe have R B,A ( λ ) = ( B − λ ) − − ( A − λ ) − = − γ ( λ ) M ( λ ) − γ ( λ ) ∗ (2.25)by Proposition 2.7 (iii). Assume that R A,B ( λ ) ≥ R A,B ( λ ) f, f ) = (cid:0) − M ( λ ) − γ ( λ ) ∗ f, γ ( λ ) ∗ f (cid:1) ≥ , f ∈ H , and since ran( γ ( λ ) ∗ ) is dense in H (see Section 2.1) we have − M ( λ ) − ≥ T ( λ ) := γ ( λ )( − M ( λ )) − / ∈ B ( H , H ) and using the assumption (2.23)for some, and hence for all, z ∈ ρ ( A ) ∩ ρ ( B ) we conclude from (2.25) that R B,A ( λ ) = T ( λ ) T ( λ ) ∗ ∈ S p ( H ) . This relation yields T ( λ ) ∗ ∈ S p ( H , H ) and T ( λ ) ∈ S p ( H , H ), and hence γ ( λ ) = T ( λ )( − M ( λ )) / ∈ S p ( H , H ). It then follows from (2.5) that γ ( z ) ∈ S p ( H , H ) and γ ( ξ ) ∗ ∈ S p ( H , H ) , z, ξ ∈ ρ ( A ) . Combining this with (2.7) implies M ( z ) − M ( λ ) ∈ S p ( H ). Therefore, setting C := M ( λ ) and K ( z ) := M ( z ) − M ( λ ), z ∈ C + , we arrive at the representation (2.9).Note that C = M ( λ ) is a boundedly invertible self-adjoint operator. Furthermore,since Im K ( z ) = Im M ( z ) and M ( · ) ∈ R s [ H ] we conclude K ( · ) ∈ R s [ H ], that is,the Weyl function M ( · ) is S p -regular. (cid:3) Remark . Condition (2.24) is satisfied if the symmetric operator S = A ∩ B is semibounded from below and A is chosen to be its Friedrichs extension. In thiscase (2.23) yields the semiboundedness of the operator B and the inequality (2.24)holds for any λ smaller than the lower bound of B . Remark . The density of dom( A )+dom( B ) in H under the conditions of Propo-sition 2.9 is well known (see for instance [36]). The simple proof presented here andwhich does not exploit the second Neumann formula seems to be new. Remark . Proposition 2.7(i) can also be viewed as an immediate consequencefrom the fact that the values of M − ( · ) are in B ( H ); cf. [34, 38]. For the convenienceof the reader we have presented a simple direct proof.3. A representation of the scattering matrix
Let A and B be self-adjoint operators in a Hilbert space H and assume that theyare resolvent comparable, i.e. their resolvent difference is a trace class operator,( B − i ) − − ( A − i ) − ∈ S ( H ) . (3.1)Denote by H ac ( A ) the absolutely continuous subspace of A and let P ac ( A ) be theorthogonal projection in H onto H ac ( A ). In accordance with the Birman-Kreintheorem, under the assumption (3.1) the wave operators W ± ( A, B ) := s − lim t →±∞ e itB e − itA P ac ( A )exist and are complete, i.e. the ranges of W ± ( B, A ) coincide with the absolutelycontinuous subspace H ac ( B ) of B ; cf. [12, 59, 79, 81, 82]. The scattering operator S ( A, B ) of the scattering system is defined by S ( A, B ) = W + ( A, B ) ∗ W − ( A, B ) . CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 13
The operator S ( A, B ) commutes with A and is unitary in H ac ( A ), hence it is uni-tarily equivalent to a multiplication operator induced by a family { S ( A, B ; λ ) } λ ∈ R of unitary operators in a spectral representation of the absolutely continuous part A ac of A , A ac := A ↾ dom( A ) ∩ H ac ( A ) . The family { S ( A, B ; λ ) } λ ∈ R is called the scattering matrix of the scattering system { A, B } .In Theorem 3.1 and Corollary 3.3 below we shall provide a representation ofthe scattering matrix { S ( A, B ; λ ) } λ ∈ R of the system { A, B } in an extension theoryframework using B -generalized boundary triples and their Weyl functions. It isassumed that the closed symmetric operator S = A ∩ B is densely defined; in themore general framework of non-densely defined symmetric operators this assump-tion can be dropped. First we discuss the case that S = A ∩ B is simple, i.e. S does not contain a self-adjoint part or, equivalently, the condition H = clsp (cid:8) ker( S ∗ − z ) : z ∈ C \ R (cid:9) is satisfied; cf. [60]. In the sequel the abbreviation a.e. means ”almost everywherewith respect to the Lebesgue measure”. Theorem 3.1.
Let A and B be self-adjoint operators in a Hilbert space H , assumethat the closed symmetric operator S = A ∩ B is densely defined and simple, andlet Π = {H , Γ , Γ } be a B -generalized boundary triple for S ∗ such that A = T ↾ ker(Γ ) and B = T ↾ ker(Γ ) . Assume, in addition, that the Weyl function M ( · ) corresponding to Π is S -regular.Then { A, B } is a complete scattering system and L ( R , dλ, H λ ) , H λ := ran(Im M ( λ + i , forms a spectral representation of A ac such that for a.e. λ ∈ R the scattering matrix { S ( A, B ; λ ) } λ ∈ R of the scattering system { A, B } admits the representation S ( A, B ; λ ) = I H λ − i p Im M ( λ + i M ( λ + i − p Im M ( λ + i . Proof.
The proof of Theorem 3.1 consists of three separate steps and is essentiallybased on Theorem A.2. Parts of the proof follow the lines in [20, Proof of Theorem3.1], where the special case of a symmetric operator S with finite deficiency indiceswas treated.First of all we note that the S -regularity assumption on M ( · ) together withProposition 2.7 (iii) ensures that the resolvent difference of A and B is a trace classoperator. Hence the wave operators W ± ( A, B ) exist and are complete and { A, B } is a complete scattering system, see, e.g. [82, Theorem VI.5.1]. Step 1.
According to Proposition 2.7 (iii) the resolvent difference of A and B in(3.1) can be written in a Krein type resolvent formula of the form( B − z ) − − ( A − z ) − = − γ ( z ) M ( z ) − γ (¯ z ) ∗ , z ∈ ρ ( A ) ∩ ρ ( B ) . (3.2)In particular, from (3.2) and (2.5) we get( B − i ) − − ( A − i ) − = − γ ( i ) M ( i ) − γ ( − i ) ∗ = − ( A + i )( A − i ) − γ ( − i ) M ( i ) − γ ( − i ) ∗ = φ ( A ) CGC ∗ where φ ( t ) := t + it − i , t ∈ R , C := γ ( − i ) and G := − M ( i ) − . (3.3) We claim that the condition H ac ( A ) = clsp (cid:8) E acA ( δ ) ran C : δ ∈ B ( R ) (cid:9) (3.4)in Theorem A.2 is satisfied. In fact, since S is assumed to be simple we have H = clsp (cid:8) ker( S ∗ − z ) : z ∈ C \ R (cid:9) . Furthermore, using ker( S ∗ − z ) = ker( T − z ), z ∈ C \ R , which follows from (2.4),and ran( γ ( z )) = ker( T − z ), z ∈ C \ R , it follows that H = clsp (cid:8) ker( T − z ) : z ∈ C \ R (cid:9) = clsp (cid:8) γ ( z ) h : z ∈ C \ R , h ∈ H (cid:9) = clsp (cid:8) ( A + i )( A − z ) − γ ( − i ) h : z ∈ C \ R , h ∈ H (cid:9) = clsp (cid:8) ( A + i )( A − z ) − Ch : z ∈ C \ R , h ∈ H (cid:9) = clsp (cid:8) E A ( δ ) Ch : h ∈ H , δ ∈ B ( R ) (cid:9) and hence H ac ( A ) = clsp (cid:8) P ac ( A ) E A ( δ ) Ch : h ∈ H , δ ∈ B ( R ) (cid:9) . Since E acA ( δ ) = P ac ( A ) E A ( δ ) this implies (3.4). Step 2.
Now we apply Theorem A.2 to obtain a preliminary form of the scat-tering matrix { S ( A, B ; λ ) } λ ∈ R . Since M ( · ) is S -regular by assumption we haveIm M ( i ) = γ ( i ) ∗ γ ( i ) ∈ S ( H ) (see (2.8)) and hence γ ( i ) ∈ S ( H , H ) and C = γ ( − i ) = (cid:0) I − i ( A + i ) − (cid:1) γ ( i ) ∈ S ( H , H ) . Therefore the function λ C ∗ E A (( −∞ , λ )) C is S ( H )-valued and in accordancewith [25, Lemma 2.2] this function is S ( H )-differentiable for a.e. λ ∈ R . Wecompute its derivative λ K ( λ ) = ddλ C ∗ E A (( −∞ , λ )) C and the square root λ p K ( λ ) for a.e. λ ∈ R . First we note that by the S ( H )-generalization of the Fatou theorem (see [25, Lemma 2.4]) K ( λ ) = lim ε → πi C ∗ (cid:0) ( A − λ − iε ) − − ( A − λ + iε ) − (cid:1) C = lim ε → επ C ∗ (cid:0) ( A − λ − iε ) − ( A − λ + iε ) − (cid:1) C (3.5)for a.e. λ ∈ R . On the other hand, inserting formula γ ( λ + iε ) = ( A + i )( A − λ − iε ) − γ ( − i ) = ( A + i )( A − λ − iε ) − C (see (2.5)) into (2.8) givesIm M ( λ + iε ) = εγ ( λ + iε ) ∗ γ ( λ + iε )= εC ∗ ( I + A ) (cid:0) A − λ + iε (cid:1) − (cid:0) A − λ − iε (cid:1) − C. Combining this relation with (3.5) impliesIm M ( λ + i
0) = lim ε → Im M ( λ + iε ) = π (1 + λ ) K ( λ )for a.e. λ ∈ R . In particular, ran(Im M ( λ + i K ( λ )) for a.e. λ ∈ R andhence H λ = ran (cid:0) Im M ( λ + i (cid:1) = ran( K ( λ )) for a.e. λ ∈ R . CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 15
Therefore L ( R , dλ, H λ ) is a spectral representation of A ac and in accordance withTheorem A.2 the scattering matrix { S ( A, B ; λ ) } λ ∈ R is given by S ( A, B ; λ ) = I H λ + 2 πi (1 + λ ) p K ( λ ) Z ( λ ) p K ( λ )= I H λ + 2 i (1 + λ ) p Im M ( λ + i Z ( λ ) p Im M ( λ + i
0) (3.6)for a.e. λ ∈ R , where Z ( · ) is given by (A.6), Z ( λ ) = 1 λ + i Q ∗ Q + 1( λ + i ) φ ( λ ) G + lim ε → Q ∗ (cid:0) B − ( λ + iε ) (cid:1) − Q, (3.7)and Q = φ ( A ) CG = − ( A + i )( A − i ) − γ ( − i ) M ( i ) − = − γ ( i ) M ( i ) − ∈ S ( H , H ) . Observe that due to the last inclusion the limit in (3.7) exists for a.e. λ ∈ R inevery S p -norm with p > Z ( · ) in (3.7) is welldefined a.e. on R ; cf. Lemma 2.6. Step 3.
In the third and final step we prove that Z ( λ ) = −
11 + λ M ( λ + i − (3.8)for a.e. λ ∈ R . Then inserting this expression in (3.6) one arrives at the assertedform of the scattering matrix.Applying the mapping Γ to (3.2) and using ker(Γ ) = dom( A ) and Definition 2.3one getsΓ ( B − z ) − = Γ ( A − z ) − − Γ γ ( z ) M ( z ) − γ (¯ z ) ∗ = − M ( z ) − γ (¯ z ) ∗ (3.9)for z ∈ ρ ( A ) ∩ ρ ( B ) and henceΓ ( B + i ) − = − M ( − i ) − γ ( i ) ∗ = (cid:0) − γ ( i ) M ( i ) − (cid:1) ∗ = Q ∗ . This yields Q ∗ ( B − z ) − Q = Γ ( B + i ) − ( B − z ) − Q = Γ (cid:0) Q ∗ ( B − ¯ z ) − ( B − i ) − (cid:1) ∗ = Γ (cid:0) Γ ( B + i ) − ( B − ¯ z ) − ( B − i ) − (cid:1) ∗ . (3.10)In order to compute this expression we note that( B + i ) − ( B − ¯ z ) − ( B − i ) − = −
11 + ¯ z (cid:0) ( B + i ) − − ( B − ¯ z ) − (cid:1) + 12 i (¯ z − i ) (cid:0) ( B + i ) − − ( B − i ) − (cid:1) and hence (3.9) impliesΓ ( B + i ) − ( B − ¯ z ) − ( B − i ) − = 11 + ¯ z (cid:0) M ( − i ) − γ ( i ) ∗ − M (¯ z ) − γ ( z ) ∗ (cid:1) − i (¯ z − i ) (cid:0) M ( − i ) − γ ( i ) ∗ − M ( i ) − γ ( − i ) ∗ (cid:1) . Taking into account that ( M (¯ µ ) − ) ∗ = M ( µ ) − for µ ∈ ρ ( A ) ∩ ρ ( B ) we obtain forthe adjoint (cid:0) Γ ( B + i ) − ( B − ¯ z ) − ( B − i ) − (cid:1) ∗ = 11 + z (cid:0) γ ( i ) M ( i ) − − γ ( z ) M ( z ) − (cid:1) + 12 i ( z + i ) (cid:0) γ ( i ) M ( i ) − − γ ( − i ) M ( − i ) − (cid:1) . In turn, combining this identity with (3.10) yields Q ∗ ( B − z ) − Qh = Γ (cid:0) Γ ( B + i ) − ( B − ¯ z ) − ( B − i ) − (cid:1) ∗ = 11 + z (cid:0) M ( i ) − − M ( z ) − (cid:1) + 12 i ( z + i ) (cid:0) M ( i ) − − M ( − i ) − (cid:1) for z ∈ ρ ( A ) ∩ ρ ( B ). Setting here z = λ + iε ∈ C + and passing to the limit as ε → ε → Q ∗ (cid:0) B − ( λ + iε ) (cid:1) − Q = 11 + λ (cid:0) M ( i ) − − M ( λ + i − (cid:1) + 12 i ( λ + i ) (cid:0) M ( i ) − − M ( − i ) − (cid:1) (3.11)for a.e. λ ∈ R ; note that by Lemma 2.6 the limit M ( λ + i − ∈ B ( H ) exists fora.e. λ ∈ R .Moreover, we have Q ∗ Q = (cid:0) γ ( i ) M ( i ) − (cid:1) ∗ γ ( i ) M ( i ) − = M ( − i ) − γ ( i ) ∗ γ ( i ) M ( i ) − = 12 i M ( − i ) − (cid:0) M ( i ) − M ( − i ) (cid:1) M ( i ) − = 12 i (cid:0) M ( − i ) − − M ( i ) − (cid:1) . Inserting this relation and (3.11) into (3.7) and taking notations (3.3) into accountwe obtain for a.e. λ ∈ R Z ( λ ) = 1 λ + i Q ∗ Q + 1( λ + i ) φ ( λ ) G + Q ∗ (cid:0) B − ( λ + i (cid:1) − Q = 12 i ( λ + i ) (cid:0) M ( − i ) − − M ( i ) − (cid:1) −
11 + λ M ( i ) − + 11 + λ (cid:0) M ( i ) − − M ( λ + i − (cid:1) + 12 i ( λ + i ) (cid:0) M ( i ) − − M ( − i ) − (cid:1) = −
11 + λ M ( λ + i − , that is, (3.8) holds. (cid:3) Remark . Instead of the assumption that the Weyl function is S -regular onemay assume in Theorem 3.1 that R B,A ( z ) = ( B − z ) − − ( A − z ) − ∈ S ( H ) holdsfor some z ∈ ρ ( A ) ∩ ρ ( B ) and R B,A ( λ ) ≥ λ ∈ R ∩ ρ ( A ) ∩ ρ ( B ); cf.Proposition 2.10.Our next task is to drop the assumption of the simplicity of S in Theorem 3.1.If S = A ∩ B is not simple then the Hilbert space H admits an orthogonal decom-position H = H ⊕ H ′ with H = { } such that S = S ⊕ S ′ , (3.12) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 17 where S is a self-adjoint operator in the Hilbert space H and S ′ is a simplesymmetric operator in the Hilbert space H ′ ; cf. [60]. It follows that there existself-adjoint extensions A ′ and B ′ of S ′ in H ′ such that A = S ⊕ A ′ and B = S ⊕ B ′ . By restricting the boundary maps of a B -generalized boundary triple for S ∗ oneobtains a B -generalized boundary triple for the operator ( S ′ ) ∗ with the same Weylfunction. Applying Theorem 3.1 to the pair { A ′ , B ′ } yields the following variant ofTheorem 3.1; cf. [20, Proof of Theorem 3.2] for the same argument in the specialcase of finite rank perturbations. Corollary 3.3.
Let A and B be self-adjoint operators in a Hilbert space H , assumethat the closed symmetric operator S = A ∩ B is densely defined and decomposed in S = S ⊕ S ′ as in (3.12) , and let L ( R , dλ, G λ ) be a spectral representation of S ac .Let Π = {H , Γ , Γ } be a B -generalized boundary triple for S ∗ as in Theorem such that the corresponding Weyl function M ( · ) is S -regular.Then { A, B } is a complete scattering system and L ( R , dλ, H λ ⊕ G λ ) , H λ := ran(Im M ( λ + i , forms a spectral representation of A ac such that for a.e. λ ∈ R the scattering matrix { S ( A, B ; λ ) } λ ∈ R of the scattering system { A, B } admits the representation S ( A, B ; λ ) = (cid:18) S ( A ′ , B ′ ; λ ) 00 I G λ (cid:19) , where S ( A ′ , B ′ ; λ ) = I H λ − i p Im M ( λ + i M ( λ + i − p Im M ( λ + i . Scattering matrices for Schr¨odinger operators on exteriordomains
Our main objective in this section is to derive representations of the scatteringmatrices for pairs of self-adjoint Schr¨odinger operators with Dirichlet, Neumann andRobin boundary conditions on unbounded domains with smooth compact bound-aries in terms of Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. After somenecessary preliminaries in Sections 4.1 and 4.2 we formulate and prove our mainresults Theorem 4.3 and Theorem 4.7 in Sections 4.3 and 4.4, respectively. Boththeorems follow in a similar way from our general result Theorem 3.1 by fixing asuitable B -generalized boundary triple and verifying that the corresponding Weylfunction is S -regular. We also mention that along the way we obtain classicalresults on singular value estimates of resolvent differences due to Birman, Grubband others without any extra efforts; cf. Remarks 4.4 and 4.8.4.1. Preliminaries on Sobolev spaces, trace maps, and Green’s secondidentity.
Let Ω ⊂ R n be an exterior domain, that is, R n \ Ω is bounded andassume that the boundary ∂ Ω of Ω is C ∞ -smooth. We denote by H s (Ω), s ∈ R ,the usual L -based Sobolev spaces on the unbounded exterior domain Ω, and by H r ( ∂ Ω), r ∈ R , the corresponding Sobolev spaces on the compact C ∞ -boundary ∂ Ω. The corresponding scalar products will be denoted by ( · , · ), and sometimes thespace is used as an index. Recall that the Dirichlet and Neumann trace operators γ D and γ N , originallydefined as linear mappings from C ∞ (Ω) to C ∞ ( ∂ Ω), admit continuous extensionsonto H (Ω) such that the mapping (cid:18) γ D γ N (cid:19) : H (Ω) → H / ( ∂ Ω) × H / ( ∂ Ω) (4.1)is surjective. The spaces H s ∆ (Ω) = (cid:8) f ∈ H s (Ω) : ∆ f ∈ L (Ω) (cid:9) , s ∈ [0 , , (4.2)equipped with the Hilbert scalar products( f, g ) H s ∆ (Ω) = ( f, g ) H s (Ω) + (∆ f, ∆ g ) L (Ω) , f, g ∈ H s ∆ (Ω) , (4.3)will play an important role. In particular, we will use that the Dirichlet traceoperator can be extended by continuity to surjective mappings γ D : H / (Ω) → H ( ∂ Ω) and γ D : H (Ω) → H / ( ∂ Ω) , (4.4)and the Neumann trace operator can be extended by continuity to surjective map-pings γ N : H / (Ω) → L ( ∂ Ω) and γ N : H (Ω) → H − / ( ∂ Ω); (4.5)cf. [62, Theorems 7.3 and 7.4, Chapter 2] for the case of a bounded smooth domainand, e.g. [49, Lemma 3.1 and Lemma 3.2]. At the same time the second Green’sidentity( − ∆ f, g ) L (Ω) − ( f, − ∆ g ) L (Ω) = ( γ D f, γ N g ) L ( ∂ Ω) − ( γ N f, γ D g ) L ( ∂ Ω) , (4.6)well known for f, g ∈ H (Ω), remains valid for f, g ∈ H / (Ω) and extends furtherto functions f, g ∈ H (Ω)( − ∆ f, g ) L (Ω) − ( f, − ∆ g ) L (Ω) = h γ D f, γ N g i − h γ N f, γ D g i , (4.7)where h· , ·i denotes the extension of the L ( ∂ Ω)-inner product onto the dual pair H / ( ∂ Ω) × H − / ( ∂ Ω) and H − / ( ∂ Ω) × H / ( ∂ Ω), respectively. As usual, here H / ( ∂ Ω) ֒ → L ( ∂ Ω) ֒ → H − / ( ∂ Ω) (4.8)is viewed as a rigging of Hilbert spaces, that is, some uniformly positive self-adjointoperator in L ( ∂ Ω) with dom( ) = H / ( ∂ Ω) is fixed and viewed as an isomor-phism : H / ( ∂ Ω) −→ L ( ∂ Ω) . (4.9)As scalar product on H / ( ∂ Ω) we choose ( ϕ, ψ ) H / ( ∂ Ω) := ( ϕ, ψ ) L ( ∂ Ω) ; it fol-lows that H − / ( ∂ Ω) coincides with the completion of L ( ∂ Ω) with respect to( − · , − · ) L ( ∂ Ω) , and − admits an extension to an isomorphism g − : H − / ( ∂ Ω) −→ L ( ∂ Ω) . The inner product h· , ·i on the right hand side of (4.7) is h ϕ, ψ i := (cid:0) ϕ, g − ψ (cid:1) L ( ∂ Ω) , ϕ ∈ H / ( ∂ Ω) , ψ ∈ H − / ( ∂ Ω) , (4.10)and extends the L ( ∂ Ω) scalar product in the sense that h ϕ, ψ i = ( ϕ, ψ ) L ( ∂ Ω) for ϕ ∈ H / ( ∂ Ω) and ψ ∈ L ( ∂ Ω). A standard and convenient choice for in (4.9) inmany situations is ∆ := ( − ∆ ∂ Ω + I ) / : H / ( ∂ Ω) −→ L ( ∂ Ω) , (4.11) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 19 where − ∆ ∂ Ω denotes the Laplace-Beltrami operator in L ( ∂ Ω); cf. Remark 4.5 forother natural choices of . Note in this connection that ∆ maps H s ( ∂ Ω) isomor-phically onto H s − / ( ∂ Ω) for any s ∈ R .In this context we also recall the following lemma, which is essentially a conse-quence of the asymptotics of the eigenvalues of the LaplaceBeltrami operator oncompact manifolds; cf. [4, Proof of Proposition 5.4.1], [5, Theorem 2.1.2], and [17,Lemma 4.7]. Lemma 4.1.
Let K be a Hilbert space and assume that X ∈ B ( K , H s ( ∂ Ω)) has theproperty ran X ⊂ H r ( ∂ Ω) for some r > s ≥ . Then X ∈ S n − r − s (cid:0) K , H s ( ∂ Ω) (cid:1) and hence X ∈ S p ( K , H s ( ∂ Ω)) for p > n − r − s . As a useful consequence of Lemma 4.1 we note that for r > ι r : H r ( ∂ Ω) −→ L ( ∂ Ω) and ι − r : L ( ∂ Ω) −→ H − r ( ∂ Ω) satisfy ι r ∈ S n − r (cid:0) H r ( ∂ Ω) , L ( ∂ Ω) (cid:1) and ι − r ∈ S n − r (cid:0) L ( ∂ Ω) , H − r ( ∂ Ω) (cid:1) , respectively. In fact, the assertion for ι r follows after fixing a unitary operator U : L ( ∂ Ω) −→ H r ( ∂ Ω), applying Lemma 4.1 to the operator X = ι r U andnoting that the singular values of X and ι r are the same. Since the dual operator ι ′ r : L ( ∂ Ω) −→ H − r ( ∂ Ω) coincides with the canonical embedding ι − r of L ( ∂ Ω)into H − r ( ∂ Ω) the second assertion follows. By composition and (1.4) we alsoconclude ι − r ◦ ι r ∈ S n − r (cid:0) H r ( ∂ Ω) , H − r ( ∂ Ω) (cid:1) . (4.12)4.2. Schr¨odinger operators with Dirichlet, Neumann, and Robin bound-ary conditions.
Let Ω ⊂ R n be an exterior domain as in Section 4.1. In thefollowing we consider a Schr¨odinger differential expression with a bounded, mea-surable, real valued potential V , L = − ∆ + V, V ∈ L ∞ (Ω) . (4.13)With the differential expression in (4.13) one naturally associates the minimal op-erator S min f = L f, dom( S min ) = H (Ω) = (cid:8) f ∈ H (Ω) : γ D f = γ N f = 0 (cid:9) , (4.14)and the maximal operator S max f = L f, dom( S max ) = (cid:8) f ∈ L (Ω) : − ∆ f + V f ∈ L (Ω) (cid:9) , in L (Ω); the expression ∆ f in dom( S max ) is understood in the sense of distri-butions. We note that dom( S max ) equipped with the graph norm coincides withthe Hilbert space H (Ω) introduced above. In the next lemma we collect somewell-known properties of S min and S max ; for the simplicity of S we refer to [22,Proposition 2.2] and the density of H s ∆ (Ω) in dom( S ∗ ) equipped with the graphnorm is shown (for the case of a bounded domain) in [62, Chapter 2,Theorem 6.4]. Lemma 4.2.
The operator S := S min is a densely defined, closed, simple, symmet-ric operator in L (Ω) . The deficiency indices of S coincide and are both infinite, dim (cid:0) ran( S − i ) ⊥ (cid:1) = dim (cid:0) ran( S + i ) ⊥ (cid:1) = ∞ . The adjoint of the minimal operator is the maximal operator, S ∗ = S ∗ min = S max and S = S min = S ∗ max , and the spaces H s ∆ (Ω) , s ∈ [0 , , are dense in dom( S ∗ ) equipped with the graphnorm. In Sections 4.3 and 4.4 we are interested in scattering systems consisting of differ-ent self-adjoint realizations of L in L (Ω). The self-adjoint Dirichlet and Neumannoperators associated to the densely defined, semibounded, closed quadratic forms a D [ f, g ] = ( ∇ f, ∇ g ) ( L (Ω)) n + ( V f, g ) L (Ω) , dom( a D ) = H (Ω) , a N [ f, g ] = ( ∇ f, ∇ g ) ( L (Ω)) n + ( V f, g ) L (Ω) , dom( a N ) = H (Ω) , are given by A D f = L f, dom( A D ) = (cid:8) f ∈ H (Ω) : γ D f = 0 (cid:9) ,A N f = L f, dom( A N ) = (cid:8) f ∈ H (Ω) : γ N f = 0 (cid:9) , (4.15)and for a real valued function α ∈ L ∞ ( ∂ Ω) the quadratic form a α [ f, g ] = a N [ f, g ] − ( αγ D f, γ D g ) L ( ∂ Ω) , dom( a α ) = H (Ω) , is also densely defined, closed and semibounded from below, and hence gives riseto a semibounded self-adjoint operator in L (Ω), which has the form A α f = L f, dom( A α ) = (cid:8) f ∈ H / (Ω) : αγ D f = γ N f (cid:9) . (4.16)We remark that the H -regularity of the functions in dom( A D ) and dom( A N ) isa classical fact (see the monographs [3, 61, 62]) and the H / -regularity of thefunctions in dom( A α ) can be found in, e.g. [14, Corollary 6.25]; in the case thatthe coefficient α in the Robin boundary condition is continuously differentiable alsodom( A α ) is contained in H (Ω); cf. [68, Theorem 4.18].4.3. Scattering matrix for the Dirichlet and Robin realization.
In this sub-section we consider the pair { A D , A α } consisting of the self-adjoint Dirichlet andRobin operator associated to L in (4.15) and (4.16) on an exterior domain Ω ⊂ R ;here we restrict ourselves to the two dimensional situation in order to ensure thatthe trace class condition (3.1) for the resolvent difference is satisfied; cf. Remark 4.4.Before formulating and proving our main result on the system { A D , A α } werecall the definition and some useful properties of the Dirichlet-to-Neumann map.First we note that for any ψ ∈ H / ( ∂ Ω) and z ∈ ρ ( A D ) there exists a uniquesolution f z ∈ H (Ω) of the boundary value problem − ∆ f z + V f z = zf z , γ D f z = ψ ∈ H / ( ∂ Ω) . (4.17)The corresponding solution operator is given by P D ( z ) : H / ( ∂ Ω) −→ H (Ω) ⊂ L (Ω) , ψ f z . (4.18)For z ∈ ρ ( A D ) the Dirichlet-to-Neumann map Λ / ( z ) is defined byΛ / ( z ) : H / ( ∂ Ω) −→ H − / ( ∂ Ω) , ψ γ N P D ( z ) ψ, (4.19) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 21 and takes Dirichlet boundary values γ D f z of the solution f z ∈ H (Ω) of (4.17) totheir Neumann boundary values γ N f z ∈ H − / ( ∂ Ω).Now we are ready to formulate and prove a representation of the scatteringmatrix for the pair { A D , A α } . Theorem 4.3.
Let Ω ⊂ R be an exterior domain with a C ∞ -smooth boundary, let V ∈ L ∞ (Ω) and α ∈ L ∞ ( ∂ Ω) be real valued functions, and let A D and A α be theself-adjoint Dirichlet and Robin realizations of L = − ∆ + V in L (Ω) in (4.15) and (4.16) , respectively. Moreover, let Λ / ( · ) be the Dirichlet-to-Neumann mapdefined in (4.19) and let M Dα ( z ) := g − ( α − Λ / ( z )) − , z ∈ ρ ( A D ) , (4.20) where : H / ( ∂ Ω) −→ L ( ∂ Ω) denotes some uniformly positive self-adjoint oper-ator in L ( ∂ Ω) with dom( ) = H / ( ∂ Ω) as in (4.8) – (4.9) .Then { A D , A α } is a complete scattering system and L ( R , dλ, H λ ) , H λ := ran(Im M Dα ( λ + i , forms a spectral representation of A acD such that for a.e. λ ∈ R the scattering matrix { S ( A D , A α ; λ ) } λ ∈ R of the scattering system { A D , A α } admits the representation S ( A D , A α ; λ ) = I H λ − i q Im M Dα ( λ + i M Dα ( λ + i − q Im M Dα ( λ + i . Proof.
It follows from (4.15) and (4.16) that the operator A α ∩ A D coincides with theminimal operator S = L min associated with L in (4.14), which is closed, denselydefined and simple by Lemma 4.2. Define the operator T as a restriction of S ∗ tothe domain H (Ω), T f = − ∆ f + V f, dom( T ) = H (Ω) , and let Γ f := γ D f and Γ f := g − ( αγ D − γ N ) f, f ∈ dom( T ) . (4.21)We claim that Π Dα = { L ( ∂ Ω) , Γ , Γ } is a B -generalized boundary triple for S ∗ with the S -regular Weyl function M Dα ( · ) in (4.20) such that A D = T ↾ ker(Γ ) and A α = T ↾ ker(Γ ) . (4.22)In fact, for f, g ∈ dom( T ) we use (4.7) and the fact that α is real valued, andcompute (Γ f, Γ g ) − (Γ f, Γ g )= (cid:0)g − ( αγ D − γ N ) f, γ D g (cid:1) − (cid:0) γ D f, g − ( αγ D − γ N ) g (cid:1) = (cid:10) αγ D f − γ N f, γ D g (cid:11) − (cid:10) γ D f, αγ D g − γ N g i = h γ D f, γ N g i − h γ N f, γ D g i = ( T f, g ) − ( f, T g )and hence Green’s identity (2.1) is satisfied. Furthermore, the mapping γ D : dom( T ) → H / ( ∂ Ω)is well defined and surjective according to (4.4), and since : H / ( ∂ Ω) → L ( ∂ Ω)is an isomorphism we conclude ran(Γ ) = L ( ∂ Ω) , i.e., Γ is surjective. From Lemma 4.2 we directly obtain that dom( T ) = H (Ω)is dense in dom( S ∗ ) equipped with the graph norm (which is equal to the space H (Ω)) and hence we have T = S ∗ . Moreover, it follows from Green’s identity (2.1)that the restrictions T ↾ ker(Γ ) and T ↾ ker(Γ ) are both symmetric operators in L (Ω) and from the definition of the boundary maps it is clear that the self-adjointoperators A D and A α are contained in the symmetric operators T ↾ ker(Γ ) and T ↾ ker(Γ ), and hence they coincide. Therefore, Π Dα = { L ( ∂ Ω) , Γ , Γ } is a B -generalized boundary triple for S ∗ such that (4.22) holds.In order to see that the Weyl function is given by M Dα ( z ) = g − ( α − Λ / ( z )) − , z ∈ ρ ( A D ) , (4.23)we recall that Λ / ( z ) γ D f z = γ N f z for f z ∈ ker( T − z ), z ∈ ρ ( A D ), according tothe definition of the Dirichlet-to-Neumann map Λ / ( · ) in (4.19). Hence we obtain g − (cid:0) α − Λ / ( z ) (cid:1) − Γ f z = g − (cid:0) αγ D f z − Λ / ( z ) γ D f z (cid:1) = Γ f z for f z ∈ ker( T − z ) and z ∈ ρ ( A D ), and this yields (4.23) and (4.20).It remains to verify that M Dα ( · ) is S -regular. For this we denote the γ -fieldassociated to Π Dα by γ Dα ( · ) and use the relation M Dα ( z ) = M Dα ( ξ ) ∗ + ( z − ¯ ξ ) γ Dα ( ξ ) ∗ γ Dα ( z ) (4.24)(see (2.7)) with some ξ ∈ ρ ( A D ) ∩ ρ ( A α ) ∩ ρ ( A N ) ∩ R and all z ∈ ρ ( A D ). Observethat (2.6) and the choice of Γ in (4.21) yield γ Dα ( ξ ) ∗ h = Γ ( A D − ¯ ξ ) − h = − g − γ N ( A D − ¯ ξ ) − h (4.25)for all h ∈ L (Ω). Since dom( A D ) ⊂ H (Ω) we conclude from (4.1) that the rangeof the mapping γ N ( A D − ¯ ξ ) − is contained in H / ( ∂ Ω). Furthermore we have γ Dα ( ξ ) ∗ ∈ B ( L (Ω) , L ( ∂ Ω)). Then it follows from (4.25) that γ N ( A D − ¯ ξ ) − ∈ B (cid:0) L (Ω) , H − / ( ∂ Ω) (cid:1) , and, in particular, this operator is closed. But then γ N ( A D − ¯ ξ ) − is also closedwhen viewed as an operator from L (Ω) into H / ( ∂ Ω), and since this operator isdefined on the whole space L (Ω) we conclude γ N ( A D − ¯ ξ ) − ∈ B (cid:0) L (Ω) , H / ( ∂ Ω) (cid:1) . Now we use that the canonical embedding operator ι − / ◦ ι / : H / ( ∂ Ω) −→ H − / ( ∂ Ω) is compact and belongs to S ( H / ( ∂ Ω) , H − / ( ∂ Ω)) by (4.12). Thuswe have γ N ( A D − ¯ ξ ) − ∈ S (cid:0) L (Ω) , H − / ( ∂ Ω) (cid:1) and hence (4.25) yields γ Dα ( ξ ) ∗ ∈ S (cid:0) L (Ω) , L ( ∂ Ω) (cid:1) . It follows that also γ Dα ( ξ ) ∈ S ( L ( ∂ Ω) , L (Ω)) and hence for all z ∈ ρ ( A D ) γ Dα ( z ) = (cid:0) I + ( z − ξ )( A D − z ) − (cid:1) γ Dα ( ξ ) ∈ S (cid:0) L ( ∂ Ω) , L (Ω) (cid:1) . (4.26)Therefore ( z − ¯ ξ ) γ Dα ( ξ ) ∗ γ Dα ( z ) ∈ S / (cid:0) L ( ∂ Ω) (cid:1) , z ∈ ρ ( A D ) . (4.27)Since S / ( L ( ∂ Ω)) ⊂ S ( L ( ∂ Ω)) and M Dα ( ξ ) = M Dα ( ξ ) ∗ we conclude from (4.24)and (4.27) that K ( z ) := M Dα ( z ) − M Dα ( ξ ) ∈ S (cid:0) L ( ∂ Ω) (cid:1) , z ∈ C + . CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 23
Since M Dα ( · ) is a strict Nevanlinna function K ( · ) is a strict Nevanlinna function. Itremains to show that C := M Dα ( ξ ) = g − α − − g − Λ / ( ξ ) − is boundedly invertible. Using that the maps (4.4) and (4.5) are surjective and ξ ∈ ρ ( A D ) ∩ ρ ( A N ) ∩ R we find that the self-adjoint operator g − Λ / ( ξ ) − issurjective, and hence boundedly invertible in L ( ∂ Ω). From ran( α − ) ⊆ L ( ∂ Ω)we obtain that g − α − is compact and therefore M Dα ( ξ ) is a Fredholm operator.Furthermore, ker( M Dα ( ξ )) = { } as otherwise there is a non-trivial function f ξ ∈ ker( T − ξ ) with Γ f ξ = 0, so that f ξ ∈ ker( A α − ξ ). But ξ in (4.24) is also in ρ ( A α )and hence f ξ = 0; a contradiction. Thus ker( M Dα ( ξ )) = { } and hence C = M Dα ( ξ )is boundedly invertible. Therefore M Dα ( · ) is an S -regular Weyl function. Now theassertions in Theorem 4.3 follow from Theorem 3.1. (cid:3) Remark . We note that (4.26) yields γ Dα ( z ) ∗ ∈ S ( L (Ω) , L ( ∂ Ω)) for all z ∈ ρ ( A D ) and since M Dα ( z ) − ∈ B ( L ( ∂ Ω)), z ∈ ρ ( A D ) ∩ ρ ( A α ), we conclude fromKrein’s formula in Proposition 2.7 (iii) that( A α − z ) − − ( A D − z ) − = − γ Dα ( z ) M Dα ( z ) − γ Dα (¯ z ) ∗ ∈ S / ( L (Ω)) . For n = 3 , , . . . one obtains in the same way as in the proof of Theorem 4.3 using(4.12) that γ Dα ( z ) ∈ S n − (cid:0) L ( ∂ Ω) , L (Ω) (cid:1) and γ Dα ( z ) ∗ ∈ S n − (cid:0) L (Ω) , L ( ∂ Ω) (cid:1) for all z ∈ ρ ( A D ) and hence( A α − z ) − − ( A D − z ) − = − γ Dα ( z ) M Dα ( z ) − γ Dα (¯ z ) ∗ ∈ S n − ( L (Ω)) . for all z ∈ ρ ( A D ) ∩ ρ ( A α ) by Proposition 2.7 (iii). This well known result goesback to Birman [24] (see also [17, 45, 53, 54, 63] for more details on singular valueestimates in this context). Remark . There are several possibilities to choose the operator in (4.9) used forthe extension (4.10) of the L ( ∂ Ω) scalar product in the rigging (4.8). Besides thechoice ∆ = ( − ∆ ∂ Ω + I ) / in (4.11) the following choice is very convenient for thescattering matrix, since it allows to express it completely in terms of the Dirichlet-to-Neumann map: Fix some λ < min { σ ( A D ) , σ ( A N ) } and note that the restrictionΛ ( λ ) (see also the beginning of Section 5.4) of the Dirichlet-to-Neumann mapΛ / ( λ ) onto H ( ∂ Ω) is a non-negative self-adjoint operator in L ( ∂ Ω) with abounded everywhere defined inverse Λ ( λ ) − in L ( ∂ Ω); the Neumann-to-Dirichletmap. Then also the square root p Λ ( λ ) is a non-negative self-adjoint operator in L ( ∂ Ω) which is boundedly invertible, and we have dom( p Λ ( λ )) = H / ( ∂ Ω);cf. [18, Proposition 3.2 (iii)]. Hence = p Λ ( λ ) : H / ( ∂ Ω) −→ L ( ∂ Ω)is a possible choice for the definition of the scalar product h· , ·i in (4.10).Following [23, Section 1] one defines the adjoint X + of an operator X ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) in the rigging H / ( ∂ Ω) ֒ → L ( ∂ Ω) ֒ → H − / ( ∂ Ω) via h Xϕ, ψ i = h ϕ, X + ψ i , ϕ, ψ ∈ H / ( ∂ Ω) . The imaginary part of the operator X is defined by Im X = i ( X − X + ), theoperator X is self-adjoint if X = X + and X is non-negative if h Xϕ, ϕ i ≥ ϕ ∈ H / ( ∂ Ω).From the fact that the function M Dα ( · ) in (4.20) is S -regular with values in B ( L ( ∂ Ω)) we concludeΛ / ( z ) ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) , z ∈ C + . Together with Lemma 2.6 this yields the following corollary.
Corollary 4.6.
Let Ω ⊂ R be an exterior domain with a C ∞ -smooth boundary andlet Λ / ( · ) be the Dirichlet-to-Neumann map defined in (4.19) . Then the followingholds. (i) The limit Λ / ( λ + i
0) = lim ε → +0 Λ / ( λ + iε ) exists for a.e. λ ∈ R in thenorm of B ( H / ( ∂ Ω) , H − / ( ∂ Ω)) ; (ii) Λ / ( λ + i ∈ B ( H / ( ∂ Ω) , H − / ( ∂ Ω)) is boundedly invertible for a.e. λ ∈ R ; (iii) Λ / ( λ + iε ) − Λ / ( λ + i ∈ S p ( H / ( ∂ Ω) , H − / ( ∂ Ω)) for p ∈ (1 , ∞ ] , ε > and a.e. λ ∈ R , and lim ε → +0 (cid:13)(cid:13) Λ / ( λ + iε ) − Λ / ( λ + i (cid:13)(cid:13) S p ( H / ( ∂ Ω) ,H − / ( ∂ Ω)) = 0;(iv) Im Λ / ( λ + i
0) = lim ε → +0 Im Λ / ( λ + iε ) exists for a.e. λ ∈ R in the S ( H / ( ∂ Ω) , H − / ( ∂ Ω)) -norm and − Im Λ / ( λ + i > . Scattering matrix for the Neumann and Robin realization.
In this sub-section we discuss a representation of the scattering matrix for the pair { A N , A α } consisting of the self-adjoint Neumann and Robin operator associated to L in(4.15) and (4.16). Here Ω is an exterior domain in R or R ; in both situations it isknown from [15, 58] that the trace class condition (3.1) for the resolvent differenceis satisfied; cf. Remark 4.8.In a similar way as in the previous subsection we first define the Neumann-to-Dirichlet map N ( z ) as an operator in L ( ∂ Ω) for all z ∈ ρ ( A N ). Recall first thatfor ϕ ∈ L ( ∂ Ω) and z ∈ ρ ( A N ) the boundary value problem − ∆ f z + V f z = zf z , γ N f z = ϕ, (4.28)admits a unique solution f z ∈ H / (Ω). The corresponding solution operator isgiven by P N ( z ) : L ( ∂ Ω) −→ H / (Ω) ⊂ L (Ω) , ϕ f z . (4.29)For z ∈ ρ ( A N ) the Neumann-to-Dirichlet map is defined by N ( z ) : L ( ∂ Ω) −→ L ( ∂ Ω) , ϕ γ D P N ( z ) ϕ. (4.30)It is clear that N ( z ) maps Neumann boundary values γ N f z of the solutions f z ∈ H / (Ω) of (4.28) onto their Dirichlet boundary values γ D f z ; here γ N and γ D denote the extensions of the Dirichlet and Neumann trace operators onto H / (Ω)from (4.4) and (4.5), respectively. Since (4.28) admits a unique solution for each ϕ ∈ L ( ∂ Ω) it is clear that the operators P N ( z ) and N ( z ) are well defined on L ( ∂ Ω).In the next theorem the scattering matrix of the pair { A N , A α } is expressed interms of the limit values of the Neumann-to-Dirichlet map N ( z ) and the parameter CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 25 α in the boundary condition of the Robin realization A α . In contrast to Theorem 4.3here it is also assumed that α − ∈ L ∞ ( ∂ Ω).
Theorem 4.7.
Let Ω ⊂ R n , n = 2 , , be an exterior domain with a C ∞ -smoothboundary, let V ∈ L ∞ (Ω) and α ∈ L ∞ ( ∂ Ω) be real valued functions such that α − ∈ L ∞ ( ∂ Ω) , and let A N and A α be the self-adjoint Neumann and Robin realizationsof L = − ∆ + V in L (Ω) in (4.15) and (4.16) , respectively. Moreover, let N ( · ) bethe Neumann-to-Dirichlet map defined in (4.30) .Then { A N , A α } is a complete scattering system and L ( R , dλ, H λ ) , H λ := ran(Im N ( λ + i , forms a spectral representation of A acN such that for a.e. λ ∈ R the scattering matrix { S ( A N , A α ; λ ) } λ ∈ R of the scattering system { A N , A α } admits the representation S ( A N , A α ; λ ) = I H λ + 2 i p Im N ( λ + i (cid:0) I − α N ( λ + i (cid:1) − α p Im N ( λ + i . Proof.
First we note that the assumption α − ∈ L ∞ ( ∂ Ω) implies A N ∩ A α = S ,where S is the minimal operator associated to L in (4.14). Recall that S is closed,densely defined and simple by Lemma 4.2. Define the operator T as a restrictionof S ∗ by T f = − ∆ f + V f, dom( T ) = H / (Ω) , and let Γ f := γ N f and Γ f := γ D f − α γ N f, f ∈ dom( T ) . (4.31)We claim that Π Nα = { L ( ∂ Ω) , Γ , Γ } is a B -generalized boundary triple for S ∗ with the S -regular Weyl function M Nα ( z ) = N ( z ) − α , z ∈ ρ ( A N ) , (4.32)such that A N = T ↾ ker(Γ ) and A α = T ↾ ker(Γ ) . (4.33)In fact, Green’s identity (2.1) is an immediate consequence of the definition ofthe boundary mappings and (4.6), and ran Γ = L ( ∂ Ω) holds by (4.5). Moreover,dom( T ) is dense in dom( S ∗ ) with respect to the graph norm by Lemma 4.2 and A α = T ↾ ker(Γ ) is clear from (4.16). Furthermore, the self-adjoint operator A N in(4.15) is contained in T ↾ ker(Γ ) and since the latter is symmetric (a consequenceof Green’s identity (2.1)) both operators coincide, that is, (4.33) holds, and Π Nα isa B -generalized boundary triple. For f z ∈ ker( T − z ), z ∈ ρ ( A N ), we have (cid:18) N ( z ) − α (cid:19) Γ f z = N ( z ) γ N f z − α γ N f z = γ D f z − α γ N f z = Γ f z , and hence the Weyl function M Nα ( · ) corresponding to Π Nα is given by (4.32).It remains to check that the Weyl function M Nα ( · ) is S -regular. This is donein a similar way as in Theorem 4.3. Denote the γ -field associated to Π Nα by γ Nα ( · )and use M Nα ( z ) = M Nα ( ξ ) ∗ + ( z − ¯ ξ ) γ Nα ( ξ ) ∗ γ Nα ( z )with some fixed ξ ∈ ρ ( A N ) ∩ ρ ( A α ) ∩ R and all z ∈ ρ ( A N ). From (4.31), (4.15),and (4.1) we obtain γ Nα ( ξ ) ∗ h = Γ ( A N − ¯ ξ ) − h = γ D ( A N − ¯ ξ ) − h ∈ H / ( ∂ Ω) and hence Lemma 4.1 yields γ Nα ( ξ ) ∗ ∈ S n − (cid:0) L (Ω) , L ( ∂ Ω) (cid:1) (4.34)and γ Nα ( z ) ∈ S n − (cid:0) L ( ∂ Ω) , L (Ω) (cid:1) (4.35)for all z ∈ ρ ( A N ). Now (1.4) shows( z − ¯ ξ ) γ Nα ( ξ ) ∗ γ Nα ( z ) ∈ S n − (cid:0) L ( ∂ Ω) (cid:1) , z ∈ ρ ( A N ) . Since S ( n − / ( L ( ∂ Ω)) ⊂ S ( L ( ∂ Ω)) for n = 2 ,
3, and M Nα ( ξ ) = M Nα ( ξ ) ∗ weconclude that K ( z ) := M Nα ( z ) − M Nα ( ξ ) ∈ S (cid:0) L ( ∂ Ω) (cid:1) , z ∈ C + . Because M Nα ( · ) is a strict Nevanlinna function K ( · ) is strict. Let us show that C := M Nα ( ξ ) = N ( ξ ) − α is invertible. In fact, since α is a boundedly invertibleoperator and N ( ξ ) is a compact operator it follows that M Nα ( ξ ) is a Fredholmoperator. Furthermore, ker( M Nα ( ξ )) is trivial as otherwise there is a non-trivialfunction f ξ ∈ ker( T − ξ ) such that Γ f ξ = 0, that is, f ξ ∈ ker( A α − ξ ). But ξ ∈ ρ ( A N ) ∩ ρ ( A α ) ∩ R yields f ξ = 0; a contradiction. Thus ker( M Nα ( ξ )) = { } andhence C := M Nα ( ξ ) is boundedly invertible. Therefore, the Weyl function M Nα ( · ) is S -regular. Now the assertions in Theorem 4.7 follow from Theorem 3.1,Im M Nα ( z ) = Im N ( z ) , M Nα ( z ) − = − (cid:0) I − α N ( z ) (cid:1) − α, z ∈ C + , andIm M Nα ( λ + i
0) = Im N ( λ + i , M Nα ( λ + i − = − (cid:0) I − α N ( λ + i (cid:1) − α for a.e. λ ∈ R . (cid:3) Remark . From (4.34) and (4.35) one concludes in the same way as in Remark 4.4that Krein’s formula in Proposition 2.7 (iii) and the property (1.4) leads to( A α − z ) − − ( A D − z ) − = − γ Nα ( z ) M Nα ( z ) − γ Nα (¯ z ) ∗ ∈ S n − ( L (Ω));for all z ∈ ρ ( A α ) ∩ ρ ( A N ); cf. [15, 58]. Remark . The definition of the boundary triples Π Dα and Π Nα in Theorems 4.3 and4.7 given for an exterior domain Ω, and the form and properties of the correspondingWeyl functions remain the same in the case of a bounded domain Ω with smoothboundary. The constructions and properties are only based on the compactnessand smoothness of ∂ Ω.5.
Schr¨odinger operators with interactions supported onhypersurfaces
In this section we investigate scattering systems consisting of Schr¨odinger op-erators in R n . Here the Euclidean space is decomposed into a smooth boundeddomain and its complement, and the usual self-adjoint Schr¨odinger operator on thewhole space is compared with the orthogonal sum of the Dirichlet or Neumannoperators on the subdomains in Section 5.2 and 5.3, and with a Schr¨odinger oper-ator with a singular δ -potential supported on the interface in Section 5.4. In ourmain results Theorem 5.1, 5.4, and 5.6 we obtain explicit forms of the scatteringmatrices in terms of Dirichlet-to-Neumann or Neumann-to-Dirichlet maps. As inSection 4 the strategy in the proofs is to apply the general result Theorem 3.1 CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 27 to suitable B -generalized boundary triples. Here we shall assume for conveniencethat a simplicity condition for the underlying symmetric operator is satisfied; thiscondition can be dropped in which case Corollary 3.3 would yield a slightly moreinvolved representation of the scattering matrix. We also refer the interested readerto Remarks 5.2, 5.5, and 5.7, where singular value estimates due to Birman, Grubband others are revisited.5.1. Preliminaries on orthogonal sums and couplings of Schr¨odinger op-erators.
Let Ω − ⊂ R n be a bounded domain with C ∞ -smooth boundary ∂ Ω − andlet Ω + := R n \ Ω − be the corresponding C ∞ -smooth exterior domain. Denote thecommon boundary of Ω + and Ω − by C := ∂ Ω ± . Throughout this section we con-sider a Schr¨odinger differential expression with a bounded, measurable, real valuedpotential V on R n , L = − ∆ + V, V ∈ L ∞ ( R n ) . (5.1)In the following we shall adapt the notation from Section 4.1 in an obviousway, e.g. H s (Ω ± ) and H r ( C ) denote the Sobolev spaces on Ω ± and the commonboundary (or interface) C , respectively, the spaces H s ∆ (Ω ± ), s ∈ [0 , H s ∆ ( R n \ C ) := H s ∆ (Ω + ) × H s ∆ (Ω − ) , s ∈ [0 , . A function f : R n → C is often written in a two component form f = { f + , f − } ,where f ± : Ω ± → C denote the restrictions of f onto Ω ± . The Dirichlet andNeumann trace operators will be denoted by γ ± D and γ ± N , and we emphasize thatthe Neumann trace is taken with respect to the outer normal of Ω ± . In particular, γ + N f + + γ − N f − = 0 for a function f = { f + , f − } ∈ H ( R n ). We also note that themapping properties of the Dirichlet and Neumann trace operators in (4.4) and (4.5)are valid for both domains Ω + and Ω − , and the same is true for the extensions ofGreen’s identity in (4.6) and (4.7), respectively. Furthermore, we shall use in theproofs in Section 5.2 and Section 5.3 that γ ± D and γ ± N admit continuous extensions γ ± D : H (Ω ± ) → H − / ( C ) and γ ± N : H (Ω ± ) → H − / ( C )and that Green’s identity extends to f ± ∈ H (Ω ± ) and g ± ∈ H (Ω ± ) in the form( − ∆ f ± , g ± ) L (Ω ± ) − ( f ± , − ∆ g ± ) L (Ω ± ) = h γ ± D f ± , γ ± N g ± i − h γ ± N f ± , γ ± D g ± i ; (5.2)cf. [62] and [52, Chapter I, Theorem 3.3 and Corollary 3.3]. In (5.2) the innerproducts h· , ·i on the right hand side denote the continuations of the L ( C ) innerproduct onto H / ( C ) × H − / ( C ) and H / ( C ) × H − / ( C ), respectively, and inthe following it will always be clear from the context which duality is used; cf.(4.8)–(4.10).The differential expression (5.1) induces self-adjoint operators in L ( R n ). Thenatural self-adjoint realization is the free Schr¨odinger operator, A free f = L f, dom( A free ) = H ( R n ) , (5.3)which is semibounded from below. Clearly the functions in dom( A free ) do not reflectthe decomposition of R n into the domains Ω + and Ω − . Furthermore, we will makeuse of the self-adjoint orthogonal sum A D = A + D ⊕ A − D , dom( A D ) = (cid:8) f = { f + , f − } ∈ H (Ω + ) ⊕ H (Ω − ) : γ + D f + = γ − D f − = 0 (cid:9) , (5.4) of the self-adjoint Dirichlet operators A ± D in L (Ω ± ) in (4.15), and of the self-adjointorthogonal sum A N = A + N ⊕ A − N , dom( A N ) = (cid:8) f = { f + , f − } ∈ H (Ω + ) ⊕ H (Ω − ) : γ + N f + = γ − N f − = 0 (cid:9) , (5.5)of the self-adjoint Neumann operators A ± N in L (Ω ± ) in (4.15). We shall sometimesrefer to A D as Dirichlet realization of L with respect to C and to A N as Neumannrealization of L with respect to C . The properties of A ± D and A ± N extend in anatural way to their orthogonal sums A D and A N in (5.4) and (5.5), respectively.In particular, the Dirichlet realization A D and the Neumann realization A N of L with respect to C are both semibounded from below.5.2. Scattering matrix for the free Schr¨odinger operator and the Dirichletrealization with respect to C . We shall derive a representation for the scatter-ing matrix of the scattering system { A D , A free } in R . Let Λ ± / ( z ) : H / ( C ) H − / ( C ) be the Dirichlet-to-Neumann map defined in (4.19) with respect to Ω ± ,that is, Λ ± / ( z ) γ ± D f ± z = γ ± N f ± z (5.6)holds for any solution f ± z ∈ H (Ω ± ) of the equation − ∆ f ± z + V ± f ± z = zf ± z and z ∈ ρ ( A ± D ). Furthermore, define the operator-valued function Λ / ( · ) byΛ / ( z ) := Λ +1 / ( z ) + Λ − / ( z ) : H / ( C ) −→ H − / ( C ) , z ∈ ρ ( A D ) . (5.7) Theorem 5.1.
Let Ω ± ⊂ R be as above, let V ∈ L ∞ ( R ) be a real valued function,and let A free and A D be the self-adjoint Schr¨odinger operators in L ( R ) in (5.3) and (5.4) , respectively. Moreover, let Λ / ( · ) be given by (5.7) and let M D free ( z ) := − g − Λ / ( z ) − , z ∈ C + , (5.8) where : H / ( C ) −→ L ( C ) denotes some uniformly positive self-adjoint operatorin L ( C ) with dom( ) = H / ( C ) as in (4.8) – (4.9) .Then { A D , A free } is a complete scattering system. If the symmetric operator S := A D ∩ A free has no eigenvalues then L ( R , dλ, H λ ) , H λ := ran (cid:0) Im M D free ( λ + i (cid:1) , forms a spectral representation of A acD such that for a.e. λ ∈ R the scattering matrix { S ( A D , A free ; λ ) } λ ∈ R of the scattering system { A D , A free } admits the representation S ( A D , A free ; λ ) = I H λ − i q Im M D free ( λ + i M D free ( λ + i − q Im M D free ( λ + i . Proof.
The closed symmetric operator S = A D ∩ A free in L ( R ) is given by Sf = L f, dom( S ) = (cid:8) f = { f + , f − } ∈ H ( R ) : γ + D f + = γ − D f − = 0 (cid:9) . (5.9)It is clear that S is a closed extension of the orthogonal sum of the minimal operators S + ⊕ S − associated to the restriction of L onto Ω + and Ω − as in (4.14) andLemma 4.2. It follows that S is densely defined and since we have assumed that CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 29 S has no eigenvalues it follows from [21, Corollary 4.4] that S is simple. We claimthat the adjoint S ∗ is given by S ∗ f = L f, dom( S ∗ ) = (cid:8) f = { f + , f − } ∈ H ( R \ C ) : γ + D f + = γ − D f − (cid:9) . In fact, since S ∗ ⊂ ( S + ) ∗ ⊕ ( S − ) ∗ it follows thatdom( S ∗ ) ⊂ H ( R \ C ) = dom( S + ) ∗ × dom( S − ) ∗ and that S ∗ f = L f for f ∈ dom( S ∗ ). Therefore, we only have to verify that f = { f + , f − } ∈ dom( S ∗ ) satisfies the interface condition γ + D f + = γ − D f − . (5.10)Assume that for f = { f + , f − } ∈ dom( S ∗ ) and all h = { h + , h − } ∈ dom( S ) we have( Sh, f ) L ( R ) = ( h, S ∗ f ) L ( R ) , that is, ( − ∆ h + , f + ) L (Ω + ) +( − ∆ h − , f − ) L (Ω − ) = ( h + , − ∆ f + ) L (Ω + ) + ( h − , − ∆ f − ) L (Ω − ) . Then it follows from Green’s identity (5.2) and the conditions γ ± D h ± = 0 and γ + N h + + γ − N h − = 0 that0 = ( − ∆ h + , f + ) L (Ω + ) − ( h + , − ∆ f + ) L (Ω + ) + ( − ∆ h − , f − ) L (Ω − ) − ( h − , − ∆ f − ) L (Ω − ) = h γ + D h + , γ + N f + i − h γ + N h + , γ + D f + i + h γ − D h − , γ − N f − i − h γ − N h − , γ − D f − i = h γ − N h − , γ + D f + − γ − D f − i holds for all h = { h + , h − } ∈ dom( S ). This implies (5.10).Now we proceed in a similar manner as in the proofs of Theorem 4.3 and Theo-rem 4.7 in the previous section. We consider the operator T defined as a restrictionof S ∗ by T f = L f, dom( T ) = (cid:8) f = { f + , f − } ∈ H ( R \ C ) : γ + D f + = γ − D f − (cid:9) , and for f ∈ dom( T ) we agree the notation γ D f := γ + D f + = γ − D f − , f = { f + , f − } ∈ dom( T ) . (5.11)We claim that Π D free = { L ( C ) , Γ , Γ } , whereΓ f := γ D f and Γ f := − g − (cid:0) γ + N f + + γ − N f − (cid:1) , f ∈ dom( T ) , is a B -generalized boundary triple with an S -regular Weyl function given by (5.8)such that A D = T ↾ ker(Γ ) and A free = T ↾ ker(Γ ) . (5.12) In fact, for f = { f + , f − } , g = { g + , g − } ∈ dom( T ) we compute with the help ofGreen’s identity (4.7) and (4.10) that(Γ f, Γ g ) − (Γ f, Γ g )= h− γ + N f + − γ − N f − , γ D g i − h γ D f, − γ + N g + − γ − N g − i = h γ + D f + , γ + N g + i − h γ + N f + , γ + D g + i + h γ − D f − , γ − N g − i − h γ − N f − , γ − D g − i = ( − ∆ f + , g + ) − ( f + , − ∆ g + ) + ( − ∆ f − , g − ) − ( f − , − ∆ g − )= ( T f, g ) − ( f, T g )and (4.4) implies ran(Γ ) = L ( C ) in the present situation; cf. the proof of Theo-rem 4.3. Since T ↾ ker(Γ ) and T ↾ ker(Γ ) are both symmetric operators by (2.1),and contain the self-adjoint operators A D and A free , respectively, it follows that(5.12) is satisfied. Furthermore, as S = A D ∩ A free it is clear that the self-adjointoperator A D and A free are disjoint extensions of S . It follows thatdom( A D ) + dom( A free ) (5.13)is dense in dom( S ∗ ) with respect to the graph norm; cf. Proposition 2.9. Since thespace (5.13) is contained in dom( T ) ⊂ dom( S ∗ ) we conclude T = S ∗ . ThereforeΠ D free is B -generalized boundary triple such that (5.12) holds.Next we show that the Weyl function M D free ( · ) corresponding to Π D free is S -regular and has the form in (5.8). Let f z = { f + z , f − z } ∈ ker( T − z ), z ∈ ρ ( A D ), anduse (5.6) and (5.7) to compute − g − Λ / ( z ) − Γ f z = − g − (cid:0) Λ +1 / ( z ) + Λ / ( z ) − (cid:1) γ D f z = − g − ( γ + N f + z + γ − N f − z )= Γ f z . Hence the Weyl function is M D free ( z ) = − g − Λ / ( z ) − . In order to see that M D free ( · )is S -regular we proceed in the same way as in the proof of Theorem 4.3. Let γ D free ( · )be the γ -field corresponding to the B -generalized boundary triple Π D free and use M D free ( z ) = M D free ( ξ ) ∗ + ( z − ¯ ξ ) γ D free ( ξ ) ∗ γ D free ( z ) (5.14)(see (2.7)) with some ξ ∈ ρ ( A D ) ∩ ρ ( A free ) ∩ ( −∞ , ess inf V ) and all z ∈ ρ ( A D ). For h = { h + , h − } ∈ L ( R n ) we have γ D free ( ξ ) ∗ h = Γ ( A D − ¯ ξ ) − h = − g − (cid:0) γ + N ( A + D − ¯ ξ ) − h + + γ − N ( A − D − ¯ ξ ) − h − (cid:1) (5.15)and since dom( A D ) ⊂ H (Ω + ) × H (Ω − ) we conclude from (4.1) that γ + N ( A + D − ¯ ξ ) − h + + γ − N ( A − D − ¯ ξ ) − h − ∈ H / ( C ) . As in the proof of Theorem 4.3 it then follows from (4.12) that γ D free ( ξ ) ∗ ∈ S (cid:0) L ( R n ) , L ( C ) (cid:1) (5.16)and γ D free ( z ) ∈ S ( L ( C ) , L ( R n )) for all z ∈ ρ ( A D ). Hence (5.14) yields that K ( z ) := M D free ( z ) − M D free ( ξ ) ∈ S (cid:0) L ( ∂ Ω) (cid:1) , z ∈ C + , CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 31 where it was used that M D free ( ξ ) ∗ = M D free ( ξ ). Let us show that M D free ( ξ ) is boundedlyinvertible. For ξ ∈ ρ ( A D ) ∩ ρ ( A free ) ∩ ( −∞ , ess inf V ) one checks that the opera-tors g − Λ ± / ( ξ ) − are non-negative and the same considerations as in the end ofthe proof of Theorem 4.3 show that these operators are surjective and boundedlyinvertible, and hence uniformly positive. This implies that also g − (cid:0) Λ +1 / ( ξ ) + Λ − / ( ξ ) (cid:1) − is uniformly positive. Hence, M D free ( ξ ) is boundedly invertible which shows that M D free ( · ) is S -regular. Now the assertions follow directly from Theorem 3.1. (cid:3) Remark . As in Remarks 4.4 and 4.8 it follows from (5.15) and (4.12) in thesame way as in (5.16) that γ D free ( z ) ∗ ∈ S n − (cid:0) L ( R n ) , L ( C ) (cid:1) for z ∈ ρ ( A D ). This yields γ D free ( z ) ∈ S n − ( L ( C ) , L ( R n )) for z ∈ ρ ( A D ) and henceKrein’s formula in Proposition 2.7 (iii) implies( A free − z ) − − ( A D − z ) − = − γ D free ( z ) M D free ( z ) − γ D free (¯ z ) ∗ ∈ S n − ( L ( R n ));for all z ∈ ρ ( A free ) ∩ ρ ( A D ); cf. [24, 54]. For further development with applicationsto the scattering theory we also refer the reader to [33] and [79]. Remark . In a similar way as in Remark 4.5 there is a particularly convenientchoice of the operator in (4.8)–(4.9) in the present context. Namely, since for z < min { σ ( A ± D ) , σ ( A ± N ) } the self-adjoint operators q Λ +1 / ( z ) and q Λ − / ( z )defined on H / ( C ) are non-negative and boundedly invertible in L ( C ) it followsthat := q Λ +1 / ( z ) + q Λ − / ( z ) : H / ( C ) −→ L ( C )is a possible choice for the definition of the scalar product h· , ·i in (4.10).5.3. Scattering matrix for the free Schr¨odinger operator and the Neu-mann realization with respect to C . In this section we consider the pair { A N , A free } consisting of the orthogonal sum A N = A + N ⊕ A − N of the Neumannoperators in (5.5) and the free Schr¨odinger operator in (5.3). We first define theNeumann-to-Dirichlet maps N ±− / ( z ) : H − / ( C ) −→ H / ( C ) , z ∈ ρ ( A N ) , as extensions of the Neumann-to-Dirichlet maps on L ( C ) defined in the beginningof Section 4.4. More precisely, we recall that for φ ± ∈ H − / ( C ) and z ∈ ρ ( A ± N )the boundary value problem − ∆ f ± + V ± f ± = zf ± , γ ± N f ± = φ ± , (5.17)admits a unique solution f ± z ∈ H (Ω ± ). The corresponding solution operator isdenoted by P ± N ( z ) : H − / ( C ) −→ H ( C ) ⊂ L ( C ) , φ ± f ± z . Note that the restriction of P ± N ( z ) onto L ( C ) coincides with the solution operatordefined in (4.29). For z ∈ ρ ( A ± N ) the Neumann-to-Dirichlet map is defined by N ±− / ( z ) : H − / ( C ) −→ H / ( C ) , φ ± γ ± D P ± N ( z ) φ ± . (5.18)Clearly, N ±− / ( z ) is an extension of the Neumann-to-Dirichlet map defined in (4.30)onto H − / ( C ), the operators in (5.18) map Neumann boundary values γ ± N f ± z ofsolutions f ± z ∈ H (Ω ± ) of (5.17) to the corresponding Dirichlet boundary values γ ± D f ± z ∈ H / ( C ).In the next theorem we obtain an expression for the scattering matrix of the pair { A N , A free } in terms of the sum N − / ( z ) := N + − / ( z ) + N −− / ( z ) : H − / ( C ) −→ H / ( C ) , z ∈ ρ ( A N ) , (5.19)of the Neumann-to-Dirichlet maps in (5.18). Theorem 5.4.
Let Ω ± ⊂ R be as above, let V ∈ L ∞ ( R ) be a real valued function,and let A free and A N be the self-adjoint Schr¨odinger operators in L ( R ) in (5.3) and (5.5) , respectively. Moreover, let N − / ( · ) be given by (5.19) and let M N free ( z ) := N − / ( z ) e , z ∈ C + , where : H / ( C ) −→ L ( C ) denotes some uniformly positive self-adjoint operatorin L ( C ) with dom( ) = H / ( C ) as in (4.8) – (4.9) .Then { A N , A free } is a complete scattering system. If the symmetric operator S := A N ∩ A free has no eigenvalues then L ( R , dλ, H λ ) , H λ := ran (cid:0) Im M N free ( λ + i (cid:1) , forms a spectral representation of A acN such that for a.e. λ ∈ R the scattering matrix { S ( A N , A free ; λ ) } λ ∈ R of the scattering system { A N , A free } admits the representation S ( A N , A free ; λ ) = I H λ − i q Im M N free ( λ + i M N free ( λ + i − q Im M N free ( λ + i . Proof.
The proof of Theorem 5.4 is very similar to the proof of Theorem 5.1, andhence we present a sketch only. Consider the closed symmetric operator S = A N ∩ A free in L ( R ) which is given by Sf = L f, dom( S ) = (cid:8) f = { f + , f − } ∈ H ( R ) : γ + N f + = γ − N f − = 0 (cid:9) . It follows that S is densely defined, the assumption σ p ( S ) = ∅ and same argumentsas in [21, Proof of Lemma 4.3] ensure that S is simple, and a similar considerationas in the proof of Theorem 5.1 shows that the adjoint S ∗ is given by S ∗ f = L f, dom( S ∗ ) = (cid:8) f = { f + , f − } ∈ H ( R \ C ) : γ + N f + = γ − N f − (cid:9) . Next we consider the operator T defined as a restriction of S ∗ by T f = L f, dom( T ) = (cid:8) f = { f + , f − } ∈ H ( R \ C ) : γ + N f + = γ − N f − (cid:9) , and one verifies in the same way as in the proof of Theorem 5.1 that Π N free = { L ( C ) , Γ , Γ } , whereΓ f := g − γ + N f + and Γ f := (cid:0) γ + D f + − γ − D f − (cid:1) , f ∈ dom( T ) , CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 33 is a B -generalized boundary triple with the Weyl function M N free ( · ) given by (5.8)such that A N = T ↾ ker(Γ ) and A free = T ↾ ker(Γ ) . Let us show that the Weyl function M N free ( · ) is S -regular. Denote the γ -fieldcorresponding to the B -generalized boundary triple Π N free by γ N free ( · ) and use M N free ( z ) = M N free ( ξ ) ∗ + ( z − ¯ ξ ) γ N free ( ξ ) ∗ γ N free ( z ) (5.20)with some fixed ξ ∈ ρ ( A N ) ∩ ρ ( A free ) ∩ ( −∞ , ess inf V ) and all z ∈ ρ ( A N ). From(4.1) and dom( A N ) ⊂ H (Ω + ) × H (Ω − ) we conclude for h = { h + , h − } ∈ L ( R n )that − γ N free ( ξ ) ∗ h = − Γ ( A N − ¯ ξ ) − h = γ + D ( A + N − ¯ ξ ) − h + − γ − D ( A − N − ¯ ξ ) − h − ∈ H / ( C ) . (5.21)Since − γ N free ( ξ ) ∗ ∈ B ( L ( R ) , H / ( C )) Lemma 4.1 yields − γ N free ( ξ ) ∗ ∈ S (cid:0) L ( R ) , H / ( C ) (cid:1) and hence γ N free ( ξ ) ∗ ∈ S (cid:0) L ( R ) , L ( C ) (cid:1) . (5.22)Therefore γ N free ( z ) ∈ S ( L ( C ) , L ( R )) for all z ∈ ρ ( A N ). Now it follows from (5.20)that K ( z ) := M N free ( z ) − M N free ( ξ ) ∈ S (cid:0) L ( C ) (cid:1) , z ∈ C + , where we have used that M N free ( ξ ) = M N free ( ξ ) ∗ . It remains to show that M N free ( ξ )is invertible, which follows from the same reasoning as in the end of the proof ofTheorem 5.1. Hence M N free ( · ) is S -regular and the assertions of Theorem 5.4 followdirectly from Theorem 3.1. (cid:3) Remark . As in Remark 5.2 the considerations in (5.21) and (5.22) together withLemma 4.1 show γ N free ( z ) ∗ ∈ S n − (cid:0) L ( R n ) , L ( C ) (cid:1) , γ N free ( z ) ∈ S n − (cid:0) L ( C ) , L ( R n ) (cid:1) for all z ∈ ρ ( A N ). Hence( A free − z ) − − ( A N − z ) − = − γ N free ( z ) M N free ( z ) − γ N free (¯ z ) ∗ ∈ S n − ( L ( R n ));for all z ∈ ρ ( A free ) ∩ ρ ( A N ); cf. [54].5.4. Schr¨odinger operators with δ -potentials supported on hypersurfaces. In this third and last application on scattering matrices for coupled Schr¨odingeroperators we consider the pair { A free , A δ,α } , where α ∈ L ∞ ( C ) is a real valuedfunction and A δ,α is a Schr¨odinger operator with δ -potential of strength α supportedon the hypersurface C defined by A δ,α f = − ∆ f + V f, dom( A δ,α ) = (cid:26) f = (cid:18) f + f − (cid:19) ∈ H / ( R n \ C ) : γ + D f + = γ − D f − ,αγ ± D f ± = γ + N f + + γ − N f − (cid:27) . (5.23)Such type of Schr¨odinger operators with singular interactions have attracted a lotof attention in the past; cf. [39] for a survey and e.g. [16] for further references andan approach via boundary mappings closely related to the present considerations.According to [16, Theorem 3.5, Proposition 3.7, and Theorem 3.16] the operator A δ,α in (5.23) is self-adjoint in L ( R n ), semibounded from below and coincides withthe self-adjoint operator associated to the closed quadratic form a δ,α [ f, g ] = ( ∇ f, ∇ g ) + ( V f, g ) − ( αγ ± D f, γ ± D g ) L ( C ) , f, g ∈ H ( R n ) . We define the Dirichlet-to-Neumann mapsΛ ± ( z ) : H ( C ) −→ L ( C ) , z ∈ ρ ( A ± D ) , as restrictions of the Dirichlet-to-Neumann maps on H / ( C ) in (4.19); cf. Re-mark 4.5. More precisely, for φ ± ∈ H ( C ) and z ∈ ρ ( A ± D ) the boundary valueproblem − ∆ f ± + V ± f ± = zf ± , γ ± D f ± = φ ± , admits a unique solution f ± z ∈ H / (Ω ± ). The corresponding solution operatorsare denoted by P ± D ( z ) : H ( C ) −→ H / ( C ) ⊂ L ( C ) , φ ± f ± z , and it is clear that the restriction of P ± D ( z ) in (4.18) onto H ( C ) coincides with P ± D ( z ). For z ∈ ρ ( A ± D ) the Dirichlet-to-Neumann maps Λ ± ( · ) on H ( C ) are givenby Λ ± ( z ) : H ( C ) −→ L ( C ) , φ ± γ ± N P ± D ( z ) φ ± , (5.24)and by construction Λ ± ( z ) are the restrictions of the Dirichlet-to-Neumann mapsΛ ± / ( z ) in (4.19) onto H ( C ).In the next theorem we obtain an expression for the scattering matrix of the pair { A free , A δ,α } in terms of the sumΛ ( z ) := Λ +1 ( z ) + Λ − ( z ) : H ( C ) −→ L ( C ) , z ∈ ρ ( A D ) , (5.25)of the Dirichlet-to-Neumann maps in (5.24). Theorem 5.6 and its proof can beviewed as a variant of Theorem 4.7; in the same way as in Theorem 4.7 it isassumed that α − ∈ L ∞ ( C ). Theorem 5.6.
Let Ω ± ⊂ R n , n = 2 , , be as above, let V ∈ L ∞ ( R n ) and α ∈ L ∞ ( C ) be real valued functions such that α − ∈ L ∞ ( C ) , and let A free and A δ,α bethe self-adjoint realizations of the Schr¨odinger expression given by (5.3) and (5.23) ,respectively. Moreover, let Λ ( · ) be as in (5.25) .Then { A free , A δ,α } is a complete scattering system. If the symmetric operator S := A free ∩ A δ,α has no eigenvalues then L ( R , dλ, H λ ) , H λ := ran(Im(Λ ( λ + i − ) , forms a spectral representation of A ac free such that for a.e. λ ∈ R the scatteringmatrix { S ( A free , A δ,α ; λ ) } λ ∈ R of the scattering system { A free , A δ,α } admits the rep-resentation S ( A free , A δ,α ; λ )= I H λ + 2 i p Im Λ ( λ + i − (cid:0) I − α Λ ( λ + i − (cid:1) − α p Im Λ ( λ + i − . Proof.
Note first that the assumptions α − ∈ L ∞ ( C ) implies that the closed sym-metric operator S = A free ∩ A δ,α is given by Sf = L f, dom( S ) = (cid:8) f = { f + , f − } ∈ H ( R n ) : γ + D f + = γ − D f − = 0 (cid:9) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 35 and hence coincides with the symmetric operator A D ∩ A free in (5.9) (in the case n = 2). It follows from [21, Corollary 4.4] that the operator S is simple and as inthe proof of Theorem 5.1 one verifies that its adjoint S ∗ is given by S ∗ f = L f, dom( S ∗ ) = (cid:8) f = { f + , f − } ∈ H ( R n \ C ) : γ + D f + = γ − D f − (cid:9) . Next we define the operator T by T f = L f, dom( T ) = (cid:8) f = { f + , f − } ∈ H / ( R n \ C ) : γ + D f + = γ − D f − (cid:9) (5.26)and for f = { f + , f − } ∈ dom( T ) we write γ D f := γ + D f + = γ − D f − as in (5.11). Wewill show that Π free δ,α = { L ( C ) , Γ , Γ } , whereΓ f = γ + N f + + γ − N f − , f ∈ dom( T ) , and Γ f = γ D f − α (cid:0) γ + N f + + γ − N f − (cid:1) , f ∈ dom( T ) , is a B -generalized boundary triple such that A free = T ↾ ker(Γ ) and A δ,α = T ↾ ker(Γ ) , (5.27)and the corresponding Weyl function M free δ,α ( z ) := Λ ( z ) − − α , z ∈ C + , (5.28)is S -regular.In fact, for f = { f + , f − } , g = { g + , g − } ∈ dom( T ) we compute with the helpof Green’s identity (4.6) and the interface conditions γ + D f + = γ − D f − and γ + D g + = γ − D g − that(Γ f, Γ g ) − (Γ f, Γ g )= (cid:0) γ D f − α − ( γ + N f + + γ − N f − ) , γ + N g + + γ − N g − (cid:1) − (cid:0) γ + N f + + γ − N f − , γ D g − α − ( γ + N g + + γ − N g − ) (cid:1) = (cid:0) γ D f, γ + N g + + γ − N g − (cid:1) − (cid:0) γ + N f + + γ − N f − , γ D g (cid:1) = ( γ + D f + , γ + N g + ) − ( γ + N f + , γ + D g + ) + ( γ − D f − , γ − N g − ) − ( γ − N f − , γ − D g − )= ( − ∆ f + , g + ) − ( f + , − ∆ g + ) + ( − ∆ f − , g − ) − ( f − , − ∆ g − )= ( T f, g ) − ( f, T g ) , which shows (2.1). In order to show that Γ is surjective we fix some λ ∈ R suchthat λ < min { σ ( A D ) , σ ( A N ) } and we note that the direct sum decompositiondom( T ) = dom( A D ) ˙+ ker( T − λ )holds since λ ∈ ρ ( A D ). It follows from (5.26) and (4.4) that γ D maps ker( T − λ )onto H ( C ). As Λ ± ( λ ) = ( N ± ( λ )) − (cf. (4.30)) are uniformly positive self-adjoint operators in L ( C ) it follows that also Λ ( λ ) = Λ +1 ( λ ) + Λ − ( λ ) is auniformly positive self-adjoint operator in L ( C ). Let ψ ∈ L ( C ), choose ϕ ∈ H ( C )and f λ = { f + λ , f − λ } ∈ ker( T − λ ) such that Λ ( λ ) ϕ = ψ and γ D f λ = ϕ . Thenwe have Γ f λ = γ + N f + λ + γ − N f − λ = Λ ( λ ) γ D f λ = Λ ( λ ) ϕ = ψ and this implies ran(Γ ) = L ( C ). It is not difficult to check that dom( A free ) and dom( A δ,α ) are contained in ker(Γ )and ker(Γ ), respectively, and since A free and A δ,α are self-adjoint and T ↾ ker(Γ )and T ↾ ker(Γ ) are symmetric by Green’s identity (2.1) it follows that (5.27) holds.From S = A free ∩ A δ,α anddom( A free ) + dom( A δ,α ) ⊂ dom( T ) ⊂ dom( S ∗ )we conclude with the help of Proposition 2.9 that T = S ∗ . Hence Π free δ,α is a B -generalized boundary triple such that (5.27) is satisfied.In order to show that the corresponding Weyl function is given by (5.28) let f z = { f + z , f − z } ∈ ker( T − z ) and z ∈ C + . Then we haveΛ ( z ) γ D f z = Λ +1 ( z ) γ + D f + z + Λ − ( z ) − γ − D f − z = γ + N f + z + γ − N f − z = Γ f z and since ker(Λ ( z )) = { } we conclude (cid:18) Λ ( z ) − − α (cid:19) Γ f z = γ D f z − α (cid:0) γ + N f + z − γ − N f − z (cid:1) = Γ f z . This proves the representation (5.28). In order to see that the Weyl function M free δ,α ( · ) is S -regular we argue in the same way as in the previous proofs. Denotethe γ -field corresponding to the B -generalized boundary triple Π free δ,α by γ free δ,α ( · ) anduse M free δ,α ( z ) = M free δ,α ( ξ ) ∗ + ( z − ¯ ξ ) γ free δ,α ( ξ ) ∗ γ free δ,α ( z ) (5.29)with some ξ ∈ ρ ( A free ) ∩ ρ ( A δ,α ) ∩ R and all z ∈ ρ ( A free ). For h = { h + , h − } ∈ L ( R n )we have γ free δ,α ( ξ ) ∗ h = Γ ( A free − ¯ ξ ) − h = γ D ( A free − ¯ ξ ) − h ∈ H / ( C )and hence Lemma 4.1 yields γ free δ,α ( ξ ) ∗ ∈ S n − (cid:0) L ( R n ) , L ( C ) (cid:1) . (5.30)As before we conclude γ free δ,α ( z ) ∈ S n − (cid:0) L ( C ) , L ( R n ) (cid:1) , z ∈ ρ ( A free ) . (5.31)It follows from (5.29) that K ( z ) := M free δ,α ( z ) − M free δ,α ( ξ ) ∈ S (cid:0) L ( C ) (cid:1) , z ∈ C + , where M free δ,α ( ξ ) = M free δ,α ( ξ ) ∗ was used. Since the operator α is boundedly invertibleand ran(Λ ( ξ ) − ) ⊆ H ( C ) the operator M free δ,α ( ξ ) is a Fredholm operator. Furthe-more, ker( M free δ,α ( ξ )) is trivial as otherwise there is an element f ξ ∈ ker( T − ξ ) withΓ f ξ = 0, that is, f ξ ∈ ker( A δ,α − ξ ). But ξ ∈ ρ ( A δ,α ) implies f ξ = 0; a contra-diction. Hence M free δ,α ( ξ ) is invertible and it follows that M free δ,α ( · ) is S -regular for n = 2 , M free δ,α ( z ) = Im Λ ( z ) , M free δ,α ( z ) − = − (cid:0) I − α Λ ( z ) − (cid:1) − α, z ∈ C + , and Im M free δ,α ( λ + i
0) = Im Λ ( λ + i ,M free δ,α ( λ + i − = − (cid:0) I − α Λ ( λ + i − (cid:1) − α, for a.e. λ ∈ R . (cid:3) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 37
Remark . As in previous remarks it follows from (5.30)–(5.31) and Krein’s for-mula that( A δ,α − z ) − − ( A free − z ) − = − γ free δ,α ( z ) M free δ,α ( z ) − γ free δ,α (¯ z ) ∗ ∈ S n − ( L ( R n ))for all z ∈ ρ ( A free ) ∩ ρ ( A δ,α ); cf. [16]. Appendix A. Spectral representation and scattering matrix
A.1.
Spectral representations and operator spectral integrals.
Let E ( · ) bea spectral measure in the separable Hilbert space H defined on the Borel sets B ( R )of the real axis R . Further, let C be a Hilbert-Schmidt operator in H . Obviously,Σ( δ ) := C ∗ E ( δ ) C , δ ∈ B ( R ) defines a trace class valued measure on B ( R ) of finitevariation; cf. [12, Lemma 3.11] The measure admits a unique decompositionΣ( · ) = Σ s ( · ) + Σ ac ( · )into a singular measure Σ s ( · ) = C ∗ E s ( · ) C and an absolutely continuous measureΣ ac ( · ) = C ∗ E ac ( · ) C . From [12, Proposition 3.13] it follows that the trace classvalued function Σ( λ ) := C ∗ E (( −∞ , λ )) C admits a derivative K ( λ ) := ddλ Σ( λ ) > λ ∈ R with respect the Lebesgue measure dλ suchthat Σ ac ( δ ) = Z δ K ( λ ) dλ, δ ∈ B ( R ) . By H λ := ran( K ( λ )) ⊆ H we define a measurable family of subspaces in H . Theorthogonal projection P ( λ ) from H onto H λ form a measurable family of projectionswhich defines by ( P f )( λ ) := P ( λ ) f ( λ ) , f ∈ L ( R , dλ, H ) , an orthogonal projection from L ( R , dλ, H ) onto a subspace which is denoted by L ( R , dλ, H λ ). Let us assume that the closed linear span of the sets E ac ( δ ) ran( C ), δ ∈ B ( R ), coincides with H ac = E ac ( R ) H . Let(Φ E ac ( δ ) Cf )( λ ) := χ δ ( λ ) p K ( λ ) f, δ ∈ B ( R ) , f ∈ H , where χ δ ( · ) denotes the characteristic function of δ ∈ B ( R ). Obviously, we have Z k (Φ E ac ( δ ) Cf )( λ ) k H dλ = Z δ k p K ( λ ) f k H dλ = k E ac ( δ ) Cf k H . Hence Φ : H ac −→ L ( R , dλ, H λ ) defines an isometry from H ac into L ( R , dλ, H λ ).Let us show that Φ is onto L ( R , dλ, H λ ). Let g ∈ L ( R , dλ, H λ ) such that0 = (Φ E ac ( δ ) Cf, g ) = Z δ ( p K ( λ ) f, g ( λ )) H dλ for f ∈ H ac , δ ∈ B ( R ). Since δ is arbitrary we find ( p K ( λ ) f, g ( λ )) H = 0 for a.e. λ ∈ R . Hence g ( λ ) ⊥ H λ for a.e. λ ∈ R which shows g ( λ ) = 0 for a.e. λ ∈ R .Hence Φ is an isometry form H ac onto the subspace L ( R , dλ, H λ ).Obviously, we have(Φ E ac ( δ ) f )( λ ) = χ δ ( λ )(Φ f )( λ ) , δ ∈ B ( R ) , f ∈ H ac . Let A be a self-adjoint operator in H and let E A ( · ) be the corresponding spec-tral measure, i.e. A = R R λ dE A ( λ ). Then M Φ = Φ A ac where M is the naturalmultiplication operator defined by( M f )( λ ) := λf ( λ ) ,f ∈ dom( M ) := { f ∈ L ( R , dλ, H λ : λf ( λ ) ∈ L ( R , dλ, H λ } . If ϕ ( · ) : R −→ R is a bounded Borel function then ϕ ( M )Φ = Φ ϕ ( A ac ). Lemma A.1.
Let A , E A ( · ) , C and K ( λ ) be as above and assume that the absolutelycontinuous subspace H ac ( A ) satisfies the condition H ac ( A ) = clsp (cid:8) E acA ( δ ) ran( C ) : δ ∈ B ( R ) (cid:9) . Then the mapping E ac ( δ ) Cf χ δ ( λ ) p K ( λ ) f for a.e. λ ∈ R , f ∈ H , onto the dense subspace span { E acA ( δ ) ran( C ) : δ ∈ B ( R ) } of H ac ( A ) admits a uniquecontinuation to an isometric isomorphism from Φ : H ac ( A ) → L ( R , dλ, H λ ) suchthat (Φ E acA ( δ ) g )( λ ) = χ δ ( λ )(Φ g )( λ ) , g ∈ H ac ( A ) , holds for any δ ∈ B ( R ) . Let us consider operator spectral integrals of the form R R dE ac ( µ ) Cf ( λ ), whichare defined whenever f ( · ) : R −→ H is a Borel measurable function, cf. [12, Section5.2]. From [12, Proposition 5.13] we find that this integral exists if and only if R R k p K ( µ ) f ( µ ) k H dµ exists and is finite. One verifies that (cid:18) Φ Z R dE ac ( µ ) Cf ( µ ) (cid:19) ( λ ) = p K ( λ ) f ( λ ) . (A.1)A.2. Scattering.
In the following let A and B be self-adjoint operators in H , let J ∈ L ( H ) be a bounded operator such that J dom A ⊆ dom B . If V := BJ − JA, dom V := dom A, is closable and its closure is a trace class operator then the wave operators W ± ( A, B ; J ) := s − lim t →±∞ e itB Je − itA P ac ( A )exist, see [12, 71, 73]. The scattering operator S J is defined by S J ( A, B ) := W + ( A, B ; J ) ∗ W − ( A, B ; J ) . Usually the wave operators W ± ( B, A ; J ) and the scattering operator S J are notthe quantities of main interest. The objects one is more interested in are the waveoperators W ± ( A, B ) := W ± ( A, B ; I ) and S ( A, B ) := S I ( A, B ). However, if theresolvent difference of A and B is compact, then the existence of W ± ( B, A ; J )yields the existence of W ± ( B, A ) and both operators are related by W ± ( A, B ; J ) = − W ± ( A, B )( A − i ) − . In particular, this yields S J ( A, B ) = S ( A, B )( I + A ) − . (A.2)The following theorem was announced in [20, Appendix A] but not proved there.Below the complete proof of theorem is given. CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 39
Theorem A.2.
Let A and B be self-adjoint operators in the separable Hilbert space H and suppose that the resolvent difference admits the factorization S ( H ) ∋ ( B − i ) − − ( A − i ) − = φ ( A ) CGC ∗ = QC ∗ , (A.3) where C ∈ S ( H , H ) , G ∈ L ( H ) , φ ( · ) : R → R is a bounded continuous functionand Q = φ ( A ) CG . Assume that the condition H ac ( A ) = clsp (cid:8) E acA ( δ ) ran( C ) : δ ∈ B ( R ) (cid:9) (A.4) is satisfied and let K ( λ ) = ddλ C ∗ E A (( −∞ , λ )) C and H λ = ran( K ( λ )) for a.e. λ ∈ R . Then L ( R , dλ, H λ ) is a spectral representation of A ac and the scatteringmatrix { S ( A, B ; λ ) } λ ∈ R of the scattering system { A, B } has the representation S ( A, B ; λ ) = I H λ + 2 πi (1 + λ ) p K ( λ ) Z ( λ ) p K ( λ ) (A.5) for a.e. λ ∈ R , where Z ( λ ) = 1 λ + i Q ∗ Q + φ ( λ )( λ + i ) G + lim ε → +0 Q ∗ ( B − ( λ + iε )) − Q (A.6) and the limit of the last term on the right hand side exists in the Hilbert-Schmidtnorm.Proof. Consider the scattering operator S J ( A, B ) := W + ( A, B ; J ) ∗ W − ( A, B ; J ) : H ac ( A ) −→ H ac ( A ), where J := − R B ( i ) R A ( i ) and R B ( ξ ) := ( B − ξ ) − , R A ( ξ ) := ( A − ξ ) − . One easily checks that V := BJ − JA = ( B − i ) − − ( A − i ) − = φ ( A ) CGC ∗ where we have used the assumption (A.3). We note that the scattering operatorcommutes with A . From [12, Theorem 18.4] one gets the representation S J ( A, B ) − W + ( A, B ; J ) ∗ W + ( A, B ; J ) = s − lim ǫ → +0 w − lim τ → +0 (cid:26) − πi Z R dE acA ( λ ) T ( τ ; λ ) δ ǫ ( A ; λ ) P ac ( A ) (cid:27) where T ( τ ; λ ) := J ∗ V − V ∗ R B ( λ + iτ ) V. and δ ǫ ( A ; λ ) := 12 πi ( R A ( λ + iǫ ) − R A ( λ − iǫ )) = 1 π ǫ ( A − λ ) + ǫ . If condition (A.3) is satisfied, then R B ( i ) = R A ( i ) + φ ( A ) CGC ∗ = R A ( i ) + QC ∗ and we get J ∗ V = − R A ( − i ) R B ( − i ) V = − R A ( − i ) CQ ∗ V − R A ( − i ) V = − R A ( − i ) CQ ∗ V − R A ( − i ) φ ( A ) CGC ∗ . Hence we find T ( τ ; λ ) = − (cid:0) R A ( − i ) CQ ∗ Q + R A ( − i ) φ ( A ) CG + CQ ∗ R B ( λ + iτ ) Q (cid:1) C ∗ . Using (A.1) we get (cid:18) Φ Z R dE acA ( µ ) T ( τ ; µ ) δ ǫ ( A ; µ ) P ac ( A ) Ch (cid:19) ( λ ) = − p K ( λ ) Z ( τ ; λ ) C ∗ δ ǫ ( A ; λ ) P ac ( A ) Ch where Z ( τ ; λ ) := 1 λ + i Q ∗ Q + φ ( λ )( λ + i ) G + Q ∗ R B ( λ + iτ ) Q. We note that the limit Q ∗ R B ( λ + i Q := lim τ → +0 Q ∗ R B ( λ + iτ ) Q exists in theHilbert-Schmidt norm. Hence the limit Z ( λ ) := lim τ → +0 Z ( τ ; λ ) exists in theoperator norm and is given by Z ( λ ) = 1 λ + i Q ∗ Q + φ ( λ )( λ + i ) G + Q ∗ R B ( λ + i Q. This gives (cid:16) Φ n s-lim ǫ → +0 w-lim τ → +0 Z R dE acA ( µ ) T ( τ ; µ ) δ ǫ ( A ; µ ) P ac ( A ) Ch o(cid:17) ( λ )= − p K ( λ ) Z ( λ ) K ( λ ) h . By the compactness of V we get that W + ( B, A ; J ) ∗ W + ( B, A ; J ) = ( I + A ) − .Therefore we have (cid:0) Φ( W + ( A, B ; J ) ∗ W + ( A, B ; J )Φ ∗ f (cid:1) ( λ ) = (1 + λ ) − f ( λ ) . Hence Φ S J ( A, B )Φ ∗ is equal to a multiplication operator with a measurable func-tion S J ( A, B ; λ ) : H λ −→ H λ given by S J ( A, B ; λ ) := (1 + λ ) − I H λ + 2 πi p K ( λ ) Z ( λ ) p K ( λ ) . Using (A.2) we find that Φ S ( A, B )Φ ∗ is a multiplication operator induced by themeasurable function S ( A, B ; λ ) : H λ −→ H λ . Both functions S J ( A, B ; λ ) and S ( A, B ; λ ) are related by S J ( A, B ; λ ) = S ( A, B ; λ )(1 + λ ) − which yields the representation (A.5). (cid:3) Acknowledgements.
Jussi Behrndt gratefully acknowledges financial supportby the Austrian Science Fund (FWF), project P 25162-N26. We are indebted toLudvig Faddeev, Boris Pavlov, and Andrea Posilicano for useful discussions andremarks. The preparation of the paper was supported by the European ResearchCouncil via ERC-2010-AdG no 267802 (“Analysis of Multiscale Systems Driven byFunctionals”).
References [1] Abels, H., Grubb, G., Wood, I.G.: Extension theory and Kre˘ın-type resolvent formulas fornonsmooth boundary value problems. J. Funct. Anal. , 4037–4100 (2014)[2] Adamyan, V.M., Pavlov, B.S.: Zero-radius potentials and M. G. Krein’s formula for gener-alized resolvents. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova , 7–23 (1986);translation in J. Sov. Math. , 1537–1550 (1988) 15371550[3] Agmon, S.: Lectures on Elliptic Boundary Value Problems. D. Van Nostrand Co., Princeton,N.J.-Toronto-London (1965)[4] Agranovich, M.S.: Elliptic operators on closed manifolds. Encyclopaedia Math. Sci. , Par-tial differential equations, VI, 1–130 Springer, Berlin, (1990) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 41 [5] Agranovich, M.S.: Elliptic boundary problems. Encyclopaedia Math. Sci. , Partial differ-ential equations, IX, 1–144, 275–281, Springer, Berlin, (1997)[6] Albeverio, A., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in QuantumMechanics. Second edition. With an appendix by Pavel Exner. AMS Chelsea Publishing,Providence, RI (2005)[7] Albeverio, S., Kostenko, A.S., Malamud, M.M., Neidhardt, H.: Spherical Schr¨odinger opera-tors with δ -type interactions. J. Math. Phys. , 052103 24pp. (2013)[8] Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators. SolvableSchr¨odinger Type Operators. London Mathematical Society Lecture Note Series , Cam-bridge University Press, Cambridge, (2000)[9] Amrein, W.O., Pearson, D.B.: M operators: a generalisation of Weyl-Titchmarsh theory. J.Comput. Appl. Math. , 1–26 (2004)[10] Antoine, J.-P., Gesztesy, F., Shabani, J.: Exactly solvable models of sphere interactions inquantum mechanics. J. Phys. A , 3687–3712 (1987)[11] Bagraev, N.T., Mikhailova, A.B., Pavlov, B.S., Prokhorov, L.V., Yafyasov, A.M.: Parameterregime of a resonance quantum switch. Phys. Rev. B , 165308 (2005)[12] Baumg¨artel, H., Wollenberg, M.: Mathematical Scattering Theory. Operator Theory: Ad-vances and Applications , Birkh¨auser, Basel, (1983)[13] Behrndt, J., Langer, M.: Boundary value problems for elliptic partial differential operatorson bounded domains. J. Funct. Anal. , 536–565 (2007)[14] Behrndt, J., Langer, M.: Elliptic operators, Dirichlet-to-Neumann maps and quasi boundarytriples. In Operator methods for boundary value problems , London Math. Soc. Lecture NoteSer. , 121–160 Cambridge Univ. Press, Cambridge, (2012)[15] Behrndt, J., Langer, M., Lobanov, I., Lotoreichik, V., Popov, I.Yu.: A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on boundeddomains. J. Math. Anal. Appl. , 750–758 (2010)[16] Behrndt, J., Langer, M., Lotoreichik, V.: Schr¨odinger operators with δ and δ ′ -potentialssupported on hypersurfaces. Ann. Henri Poincar´e , 385–423 (2013)[17] Behrndt, J., Langer, M., Lotoreichik, V.: Spectral estimates for resolvent differences of self-adjoint elliptic operators. Integral Equations Operator Theory , 1–37 (2013)[18] Behrndt, J., Langer, M., Lotoreichik, V., Rohleder, J.: Quasi boundary triples and semi-bounded self-adjoint extensions. Proc. Roy. Soc. Edinburgh Sect. A, to appear[19] Behrndt, J., Malamud, M.M., Neidhardt, H.: Scattering matrices and Weyl functions. Proc.Lond. Math. Soc. , 568–598 (2008)[20] Behrndt, J., Malamud, M.M., Neidhardt, H.: Finite rank perturbations, scattering matricesand inverse problems. In Recent advances in operator theory in Hilbert and Krein spaces ,Operator Theory: Advances and Applications , 61–85 Birkh¨auser Verlag, Basel, (2010)[21] Behrndt, J., Rohleder, J.: Spectral analysis of self-adjoint elliptic differential operators,Dirichlet-to-Neumann maps, and abstract Weyl functions. Adv. Math. , 67–87 (2016)[23] Berezanski˘ı, Yu.M.: Selfadjoint Operators in Spaces of Functions of Infinitely many Variables.Translations of Mathematical Monographs , American Mathematical Society, Providence,RI, (1986)[24] Birman, M.ˇS.: Perturbations of the continuous spectrum of a singular elliptic operator byvarying the boundary and the boundary conditions. Vestnik Leningrad. Univ. , 543–560(1962)[27] Brasche, J.F., Exner, P., Kuperin, Yu.A., ˇSeba, P.: Schr¨odinger operators with singularinteractions. J. Math. Anal. Appl. , 112–139 (1994)[28] Brasche, J.F., Malamud, M.M., Neidhardt, H.: Weyl function and spectral properties ofself-adjoint extensions. Integral Equations Operator Theory , 264–289 (2002)[29] Brown, B.M., Grubb, G., Wood, I.G.: M -functions for closed extensions of adjoint pairs ofoperators with applications to elliptic boundary problems. Math. Nachr. , 314–347 (2009) [30] Brown, B.M., Marletta, M., Naboko, S.N., Wood, I.G.: Boundary triplets and M -functionsfor non-self-adjoint operators, with applications to elliptic PDEs and block operator matrices.J. Lond. Math. Soc. , 700–718 (2008)[31] Br¨uning, J., Geyler, V., Pankrashkin, K., Spectra of self-adjoint extensions and applicationsto solvable Schr¨odinger operators. Rev. Math. Phys. , 369–442 (1939)[33] Deift, P., Simon, B.: On the decoupling of finite singularities from the question of asymptoticcompleteness in two body quantum systems. J. Funct. Anal. , 218–238 (1976)[34] Derkach, V.A., Hassi, S., Malamud, M.M., de Snoo, H.S.V.: Boundary relations and theirWeyl families. Trans. Amer. Math. Soc. , 5351–5400 (2006)[35] Derkach, V.A., Hassi, S., Malamud, M.M., de Snoo, H.S.V.: Boundary triplets and Weylfunctions. Recent developments. In Operator methods for boundary value problems , LondonMath. Soc. Lecture Note Ser. , 161–220 Cambridge Univ. Press, Cambridge, (2012)[36] Derkach, V.A., Malamud, M.M.: Weyl function of a Hermitian operator and its connectionwith characteristic function. Preprint (1985); (see also arXiv:1503.08956)[37] Derkach, V.A., Malamud, M.M.: Generalized resolvents and the boundary value problemsfor Hermitian operators with gaps. J. Funct. Anal. , 1–95 (1991)[38] Derkach, V.A., Malamud, M.M.: The extension theory of Hermitian operators and the mo-ment problem. J. Math. Sci. , 141–242 (1995)[39] Exner, P.: Leaky quantum graphs: a review. In Analysis on graphs and its applications ,Proc. Sympos. Pure Math. , 523–564 Amer. Math. Soc., Providence, RI, (2008)[40] Exner, P., Ichinose, T.: Geometrically induced spectrum in curved leaky wires. J. Phys. A , 1439–1450 (2001)[41] Exner, P., Kondej, S.: Bound states due to a strong δ interaction supported by a curvedsurface. J. Phys. A , 443–457 (2003)[42] Exner, P., Kondej, S.: Scattering by local deformations of a straight leaky wire. J. Phys. A , 4865–4874 (2005)[43] Exner, P., Yoshitomi, K.: Asymptotics of eigenvalues of the Schr¨odinger operator with astrong δ -interaction on a loop. J. Geom. Phys. , 344–358 (2002)[44] Gesztesy, F., Makarov, K.A., Naboko, S.N.: The spectral shift operator. Operator Theory:Advances and Applications , 59–90 (1999)[45] Gesztesy, F., Malamud, M.M.: Spectral theory of elliptic operators in exterior domains.arXiv:0810.1789[46] Gesztesy, F., Mitrea, M.: Generalized Robin boundary conditions, Robin-to-Dirichlet maps,and Krein-type resolvent formulas for Schr¨odinger operators on bounded Lipschitz domains.In Perspectives in partial differential equations, harmonic analysis and applications , Proc.Sympos. Pure Math. , 105–173 Amer. Math. Soc., Providence, RI, (2008)[47] Gesztesy, F., Mitrea, M.: Nonlocal Robin Laplacians and some remarks on a paper by Filonovon eigenvalue inequalities. J. Differential Equations , 2871–2896, (2009)[48] Gesztesy, F., Mitrea, M.: Robin-to-Robin maps and Krein-type resolvent formulas forSchr¨odinger operators on bounded Lipschitz domains. Operator Theory: Advances and Ap-plications , 81–113 (2009)[49] Gesztesy, F., Mitrea, M.: A description of all self-adjoint extensions of the Laplacian andKre˘ın-type resolvent formulas on non-smooth domains. J. Anal. Math. , 53–172 (2011)[50] Gesztesy, F., Mitrea, M., Zinchenko. M.: Variations on a theme of Jost and Pais. J. Funct.Anal. , Kluwer Academic PublishersGroup, Dordrecht, (1991)[52] Grubb, G.: A characterization of the non-local boundary value problems associated with anelliptic operator. Ann. Scuola Norm. Sup. Pisa , 425–513 (1968)[53] Grubb, G.: Properties of normal boundary problems for elliptic even-order systems. Ann.Scuola Norm. Sup. Pisa Cl. Sci. , 1–61 (1974)[54] Grubb, G.: Singular Green operators and their spectral asymptotics. Duke Math. J. ,477–528 (1984)[55] Grubb, G.: Krein resolvent formulas for elliptic boundary problems in nonsmooth domains.Rend. Semin. Mat. Univ. Politec. Torino , 271–297 (2008) CATTERING MATRICES AND DIRICHLET-TO-NEUMANN MAPS 43 [56] Grubb, G.: The mixed boundary value problem, Krein resolvent formulas and spectral as-ymptotic estimates. J. Math. Anal. Appl. , 339–363 (2011)[57] Grubb, G.: Perturbation of essential spectra of exterior elliptic problems. Appl. Anal. ,103–123 (2011)[58] Grubb, G.: Spectral asymptotics for Robin problems with a discontinuous coefficient. J.Spectr. Theory , 155–177 (2011)[59] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)[60] Krein, M.G.: Basic propositions of the theory of representation of Hermitian operators withdeficiency index ( m, m ). Ukrain. Mat. Zh. , 3–66 (1949)[61] Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. AcademicPress, New York-London (1968)[62] Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications.Volume I. Springer, New York, (1972)[63] Malamud, M.M.: Spectral theory of elliptic operators in exterior domains. Russ. J. Math.Phys. , 96–125 (2010)[64] Malamud, M.M., Neidhardt, H.: Perturbation determinants for singular perturbations. Russ.J. Math. Phys. , 55–98 (2014)[65] Mantile, A., Posilicano, A., Sini, M.: Self-adjoint elliptic operators with boundary conditionson not closed hypersurfaces. J. Differential Equations , 1–55 (2016)[66] Mantile, A., Posilicano, A., Sini, M.: Limiting absorption principle, generalized eigenfunc-tion expansions and scattering matrix for Laplace operators with boundary conditions onhypersurfaces. arXiv:1605.03240[67] Marletta, M.: Eigenvalue problems on exterior domains and Dirichlet to Neumann maps. J.Comput. Appl. Math. , 367–391 (2004).[68] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Uni-versity Press, 2000.[69] Mikhailova, A.B., Pavlov, B.S., Prokhorov, L.V.: Intermediate Hamiltonian via Glazman’ssplitting and analytic perturbation for meromorphic matrix-functions. Math. Nachr. ,1376–1416 (2007)[70] Naboko, S.N.: On the boundary values of analytic operator-valued functions with a positiveimaginary part. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 55–69(1987)[71] Neidhardt, H.: Zwei-Raum-Verallgemeinerung des Theorems von Rosenblum und Kato.Math. Nachr. , 195–211 (1978)[72] Pavlov, B.S., Antoniou, I.: Jump-start in the analytic perturbation procedure for theFriedrichs model. J. Phys. A , 182–186(1978)[74] Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices , 483–506 (2008)[75] Posilicano, A.: Markovian extensions of symmetric second order elliptic differential operators.Math. Nachr. , 1848–1885 (2014)[76] Posilicano, A., Raimondi, L.: Krein’s resolvent formula for self-adjoint extensions of symmet-ric second-order elliptic differential operators. J. Phys. A , 015204, 11 pages (2009)[77] Post, O.: Boundary pairs associated with quadratic forms. Math. Nachr. , 1052–1099(2016)[78] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis.Academic Press, New York-London (1972)[79] Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Scattering Theory.Academic Press, New York-London (1979)[80] Schm¨udgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts inMathematics , Springer, Dordrecht (2012)[81] Weidmann, J.: Lineare Operatoren in Hilbertr¨aumen, Teil II. B.G. Teubner, Stuttgart, (2003)[82] Yafaev, D.R.: Mathematical Scattering Theory. Translations of Mathematical Monographs , American Mathematical Society, Providence, RI, (1992) Institut f¨ur Numerische Mathematik, Technische Universit¨at Graz, Steyrergasse 30,8010 Graz, Austria
E-mail address : [email protected] URL : Institute of Applied Mathematics and Mechanics, National Academy of Science ofUkraine, Dobrovol’s’kogo Str. 1, 84100 Slavyansk, Donetsk region, Ukraine
E-mail address : [email protected] Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117Berlin, Germany
E-mail address : [email protected] URL ::