On Eisenbud s and Wigner s R-matrix: A general approach J. Behrndt a, H. Neidhardt b, E. R. Racec c, P. N. Racec d, U. Wulf e January 8, 2007 a) Technische Universität Berlin, Institut für Mathematik, Straße des 7. Juni 36, D-0623 Berlin, Germany E-mail: behrndt@math.tu-berlin.de b) Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-07 Berlin, Germany E-mail: neidhard@wias-berlin.de c) Technische Universität Cottbus, Fakultät, Postfach 0344, D- 0303 Cottbus, Germany, and Faculty of Physics, University of Bucharest, PO Box MG-, 07725 Bucharest Magurele, Romania E-mail: roxana@physik.tu-cottbus.de d) Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-07 Berlin, Germany, and National Institute of Materials Physics, PO Box MG-7, 07725 Bucharest Magurele, Romania E-mail: racec@wias-berlin.de e) Technische Universität Cottbus, Fakultät, Postfach 0344, D- 0303 Cottbus, Germany E-mail: wulf@physik.tu-cottbus.de Abstract The main objective of this paper is to give a rigorous treatment of Wigner s and Eisenbud s R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis. 2000 Mathematical Subject Classification: 47A40, 34L25, 8U20 Keywords: scattering, scattering matrix, R-matrix, symmetric and selfadjoint operators, extension theory, boundary triplet, Weyl function, ordinary differential operators
R-MATRIX 2 Introduction The R-matrix approach to scattering was originally developed by Kapur and Peierls [2] in connection with nuclear reactions. Their ideas were improved by Wigner [40, 4] and Wigner and Eisenbud [42], where the notion of the R-matrix firstly occurred. A comprehensive overview of the R-matrix theory in nuclear physics can be found in [7, 24]. The key ideas of the R-matrix theory are rather independent from the concrete physical situation. In fact, later the R-matrix method has also found several applications in atomic and molecular physics see e.g. [6, 8]) and recently it was applied to transport problems in semiconductor nano-structures [28, 29, 30, 3, 32, 33, 43, 44, 45]. In [26, 27] an attempt was made to make the R-matrix method rigorous for elliptic differential operators, see also [34, 35] for Schrödinger operators and [36, 37] for an extension to Dirac operators. The essential idea of the R-matrix theory is to divide the whole physical system into two spatially divided subsystems which are called internal and external systems, see [40, 4, 42]. The internal system is usually related to a bounded region, while the external system is given on its complement and is, therefore, spatially infinite. The goal is to represent the scattering matrix of a certain scattering system in terms of eigenvalues and eigenfunctions of an operator corresponding to the internal system with suitable chosen selfadjoint boundary conditions at the interface between the internal and external system. This might seem a little strange at first sight since scattering is rather related to the external system than to the internal one. It is the main objective of the present paper to make a further step towards a rigorous foundation of the R-matrix method in the framework of abstract scattering theory [5], in particular, in the framework of scattering theory for open quantum systems developed in [3, 4]. This abstract approach has the advantage that any type of operators, in particular, Schrödinger or Dirac operators can be treated. We start with the direct orthogonal sum L := A T of two symmetric operators A and T with equal deficiency indices acting in the Hilbert spaces H and K, respectively. From a physical point of view the systems {A, H} and {T, K} can be regarded as incomplete internal and external systems, respectively. The system {L, L}, L := H K, is also an incomplete quantum system which is completed or closed by choosing a selfadjoint extension of L. The operator L admits several selfadjoint extensions in L. In particular, there are selfadjoint extensions of the form L 0 = A 0 T 0, where A 0 and T 0 are selfadjoint extensions of A and T in H and K, respectively. Of course, in this case the quantum system {L 0, L} decomposes into the closed internal and external system {A 0, H} and {T 0, K}, respectively, which do not interact. There are other selfadjoint extensions of L in L which are not of this structure and can be regarded as Hamiltonians of quantum systems which take into account a certain interaction of the internal and external systems {A, H} and {T, K}. In the following we choose a special self-adjoint extension L of L introduced in [9] and used in [3], see also Theorem 5., which gives the right physical Hamiltonian in applications.
R-MATRIX 3 For example, let the internal system {A, H} and external system {T, K} be given by the minimal second order differential operators A = d2 dx + v and 2 T = d2 dx + V in H = L 2 x 2 l, x r )) and K = L 2 R\x l, x r )), where x l, x r ) is a finite interval and v, V are real potentials. The extension L 0 can be chosen to be the direct sum of the selfadjoint extensions of A and T corresponding to Dirichlet boundary conditions at x l and x r. According to [3, 9] the selfadjoint extension L coincides in this case with the usual selfadjoint Schrödinger operator { L = d2 vx), x x l, x r ), + ṽ, ṽx) := dx2 V x), x R\x l, x r ), in L = L 2 R), cf. Section 6.. Let again A and T be symmetric operators with equal deficiency indices in H and K, respectively. It will be assumed that the deficiency indices of A and T are finite. Then the selfadjoint operator L is a finite rank perturbation in resolvent sense of L 0 = A 0 T 0 and therefore { L, L 0 } is a complete scattering system, i.e., the wave operators W ± L, L 0 ) := s- lim t ± eitel e itl0 P ac L 0 ) exist and map onto the absolutely continuous subspace H ac L) of L, where P ac L 0 ) is the orthogonal projection onto H ac L 0 ), cf. [2]. The scattering operator S := W + L, L 0 ) W L, L 0 ) regarded as an unitary operator in the absolutely continuous subspace H ac L 0 ) is unitarily equivalent to a multiplication operator induced by a family of unitary matrices {Sλ)} λ R in a spectral representation of the absolutely continuous part of L 0. This multiplication operator {Sλ)} λ R is called the scattering matrix of the scattering system { L, L 0 } and is one of the most important objects in mathematical scattering theory. The case that the spectrum σa 0 ) is discrete is of particular importance in physical applications, e.g., modeling of quantum transport in semiconductors. In this case the scattering matrix of { L, L 0 } is given by Sλ) = I 2i Im τλ)) Mλ) + τλ) ) Im τλ)), where M ) and τ ) are certain abstract Titchmarsh-Weyl functions corresponding to the internal and external systems, respectively, see Corollary 5.2. The R-matrix {Rλ)} λ R of { L, L 0 } is defined as the Cayley transform of the scattering matrix {Sλ)} λ R, i.e., Rλ) = ii Sλ))I + Sλ)), and the problem in the R-matrix theory is to represent {Rλ)} λ R in terms of eigenvalues and eigenfunctions of a suitable chosen closed internal system
R-MATRIX 4 {Â, H}. By the inverse Cayley transform this immediately also yields a representation of the scattering matrix by the same quantities. For Schrödinger operators the problem is usually solved by choosing appropriate selfadjoint boundary conditions at the interface between the internal and external system, in particular, Neumann boundary conditions. We show that in the abstract approach to the R-matrix theory the problem can be solved within the framework of abstract boundary triplets, which allow to characterize all selfadjoint extensions of A by abstract boundary conditions, cf. [0,, 2, 9]. It is one of our main objectives to prove that there always exists a family of closed internal systems {Aλ), H} λ R given by abstract boundary conditions connected with the function τ ), such that the R-matrix {Rλ)} λ R and the scattering matrix {Sλ)} λ R of { L, L 0 } can be expressed with the help of the eigenvalues and eigenfunctions of Aλ) for a.e. λ R, cf. Theorem 5.5. This representation requires in addition that the internal Hamiltonians Aλ) satisfy Aλ) A 0, which is always true if A 0 is the Friedrichs extension of A. Moreover, our general representation results also indicate that even for small energy ranges it is rather unusual that the R-matrix and the scattering matrix can be represented by the eigenvalues and eigenfunctions of a single λ-independent internal Hamiltonian Â. As an application again the second order differential operators A = d2 dx +v 2 and T = d2 dx + V from above are investigated and particular attention is paid 2 to the case where the potential V is a real constant. Then the family {Aλ)} λ R reduces to a single selfadjoint operator, namely, to the Schrödinger operator in L 2 x l, x r )) with Neumann boundary conditions. In general, however, this is not the case. Indeed, even in the simple case where V is constant on, x l ) and x r, ) but the constants are different, a λ-dependent family of internal Hamiltonians is required for a certain energy interval to obtain a representation of the R-matrix and the scattering matrix in terms of eigenfunctions, see Section 6.2.. The condition Aλ) A 0 is always satisfied if A 0 is chosen to be the Schrödinger operator with Dirichlet boundary conditions. Finally, we note that it is not possible to represent the R-matrix and the scattering matrix in terms of eigenfunctions of an internal Hamiltonian with Dirichlet boundary conditions. The paper is organized as follows. In Section 2 we briefly recall some basic facts on boundary triplets and associated Weyl functions corresponding to symmetric operators in Hilbert spaces. It is the aim of the simple examples from semiconductor modeling in Section 2.3 to make the reader more familiar with this efficient tool in extension and spectral theory of symmetric and selfadjoint operators. Section 3 deals with semibounded extensions and representations of Weyl functions in terms of eigenfunctions of selfadjoint extensions of a given symmetric operator. In Section 4 we prove general representation theorems for the scattering matrix and the R-matrix of a scattering system which consists of two selfadjoint extensions of the same symmetric operator. Section 5 is devoted to scattering theory in open quantum systems, and with the preparations from the previous sections we easily obtain the abovementioned representation of the R-matrix and scattering matrix of { L, L 0 } in terms of the eigenfunctions of an
R-MATRIX 5 energy dependent selfadjoint operator family. In the last section the general results are applied to scattering systems consisting of orthogonal sums of regular and singular ordinary second order differential operators. 2 Boundary triplets and Weyl functions 2. Boundary triplets Let H be a separable Hilbert space and let A be a densely defined closed symmetric operator with equal deficiency indices n ± A) = dim kera i) in H. We use the concept of boundary triplets for the description of the closed extensions of A in H, see e.g. [0,, 2, 9]. Definition 2. Let A be a densely defined closed symmetric operator in H. A triplet Π = {H, Γ 0, Γ } is called a boundary triplet for the adjoint operator A if H is a Hilbert space and Γ 0, Γ : dom A ) H are linear mappings such that the abstract Green s identity, A f, g) f, A g) = Γ f, Γ 0 g) Γ 0 f, Γ g), holds for all f, g dom A ) and the mapping Γ := Γ 0 Γ ) : dom A ) H H is surjective. We refer to [] and [2] for a detailed study of boundary triplets and recall only some important facts. First of all a boundary triplet Π = {H, Γ 0, Γ } for A always exists since the deficiency indices n ± A) of A are assumed to be equal. In this case n ± A) = dim H holds. We also note that a boundary triplet for A is not unique. In order to describe the set of closed extensions  A of A with the help of a boundary triplet Π = {H, Γ 0, Γ } for A we introduce the set CH) of closed linear relations in H, that is, the set of closed linear subspaces of H H. If Θ is a closed linear operator in H, then Θ will be identified with its graph GΘ), { ) } h Θ = GΘ) = : h dom Θ). Θh Therefore, the set of closed linear operators in H is a subset of CH). Note that Θ CH) is the graph of an operator if and only if the multivalued part mul Θ) := { h H : ) } 0 h Θ is trivial. The resolvent set ρθ) and the point, continuous and residual spectrum σ p Θ), σ c Θ) and σ r Θ) of a closed linear relation Θ are defined in a similar way as for closed linear operators, cf. [3]. Recall that the adjoint relation Θ CH) of a linear relation Θ in H is defined as { ) ) } Θ k := k : h, k) = h, k h ) for all h Θ 2.) and Θ is said to be symmetric selfadjoint) if Θ Θ resp. Θ = Θ ). We note that definition 2.) extends the usual definition of the adjoint operator. Let
R-MATRIX 6 now Θ be a selfadjoint relation in H and let P op be the orthogonal projection in H onto H op := mul Θ)) = dom Θ). Then { ) ) } x x Θ op = P op x : x Θ is a selfadjoint possibly unbounded) operator in the Hilbert space H op and Θ can be written as the direct orthogonal sum of Θ op and a pure relation Θ in the Hilbert space H := P op )H = mul Θ, { ) } 0 Θ = Θ op Θ, Θ := x : x mul Θ CH ). 2.2) With a boundary triplet Π = {H, Γ 0, Γ } for A one associates two selfadjoint extensions of A defined by A 0 := A kerγ 0 ) and A := A kerγ ). 2.3) A description of all proper symmetric, selfadjoint) extensions of A is given in the next proposition. Proposition 2.2 Let A be a densely defined closed symmetric operator in H with equal deficiency indices and let Π = {H, Γ 0, Γ } be a boundary triplet for A. Then the mapping Θ A Θ := A Γ ) Θ = A { f dom A ) : Γ 0 f, Γ f) Θ } 2.4) establishes a bijective correspondence between the set CH) and the set of closed extensions A Θ A of A. Furthermore A Θ ) = A Θ holds for any Θ CH). The extension A Θ in 2.4) is symmetric selfadjoint, dissipative, maximal dissipative) if and only if Θ is symmetric selfadjoint, dissipative, maximal dissipative). It is worth to note that the selfadjoint operator A 0 = A kerγ 0 ) in 2.3) corresponds to the pure relation Θ = { 0 h ) : h H }. Moreover, if Θ is an operator, then 2.4) can also be written in the form A Θ = A ker Γ ΘΓ 0 ), 2.5) so that, in particular A in 2.3) corresponds to Θ = 0 [H]. Here and in the following [H] stands for the space of bounded everywhere defined linear operators in H. We note that if the product ΘΓ 0 in 2.5) is interpreted in the sense of relations, then 2.5) is even true for parameters Θ with mul Θ) {0}. Later we shall often be concerned with closed simple symmetric operators. Recall that a closed symmetric operator A is said to be simple if there is no nontrivial subspace which reduces A to a selfadjoint operator. By [23] this is equivalent to H = clospan { kera λ) : λ C\R }, where clospan{ } denotes the closed linear span of a set. Note that a simple symmetric operator has no eigenvalues.
R-MATRIX 7 2.2 Weyl functions and resolvents of extensions Let again A be a densely defined closed symmetric operator in H with equal deficiency indices. A point λ C is of regular type if kera λ) = {0} and the range ran A λ) is closed. We denote the defect subspace of A at the points λ C of regular type by N λ = kera λ). The space of bounded everywhere defined linear operators mapping the Hilbert space H into H will be denoted by [H, H]. The following definition was given in [0, ]. Definition 2.3 Let A be a densely defined closed symmetric operator in H, let Π = {H, Γ 0, Γ } be a boundary triplet for A and let A 0 = A kerγ 0 ). The operator-valued functions γ ) : ρa 0 ) [H, H] and M ) : ρa 0 ) [H] defined by γλ) := Γ 0 N λ ) and Mλ) := Γ γλ), λ ρa 0 ), 2.6) are called the γ-field and the Weyl function, respectively, corresponding to the boundary triplet Π. It follows from the identity dom A ) = kerγ 0 ) +N λ, λ ρa 0 ), where as above A 0 = A kerγ 0 ), that the γ-field γ ) in 2.6) is well defined. It is easily seen that both γ ) and M ) are holomorphic on ρa 0 ), and the relations and γλ) = + λ µ)a 0 λ) ) γµ), λ, µ ρa 0 ), Mλ) Mµ) = λ µ)γµ) γλ), λ, µ ρa 0 ), 2.7) are valid see []). The identity 2.7) yields that M ) is a Nevanlinna function, that is, M ) is holomorphic on C\R, Mλ) = Mλ) for all λ C\R and Im Mλ)) is a nonnegative operator for all λ in the upper half plane C + = {λ C : Im λ) > 0}. Moreover, it follows from 2.7) that 0 ρim Mλ))) holds for all λ C\R. The following well-known theorem shows how the spectral properties of the closed extensions A Θ of A can be described with the help of the Weyl function, cf. [, 2]. Theorem 2.4 Let A be a densely defined closed symmetric operator in H and let {H, Γ 0, Γ } be a boundary triplet for A with γ-field γ and Weyl function M. Let A 0 = A kerγ 0 ) and let A Θ A be a closed extension corresponding to some Θ CH) via 2.4)-2.5). Then a point λ ρa 0 ) belongs to the resolvent set ρa Θ ) if and only if 0 ρθ Mλ)) and the formula A Θ λ) = A 0 λ) + γλ) Θ Mλ) ) γλ) 2.8) holds for all λ ρa 0 ) ρa Θ ). Moreover, λ belongs to the point spectrum σ p A Θ ), to the continuous spectrum σ c A Θ ) or to the residual spectrum σ r A Θ ) if and only if 0 σ i Θ Mλ)), i = p, c, r, respectively.
R-MATRIX 8 2.3 Regular and singular Sturm-Liouville operators We are going to illustrate the notions of boundary triplets, Weyl functions and γ-fields with some well-known simple examples. 2.3. Finite intervals Let us first consider a Schrödinger operator on the bounded interval x l, x r ) R. The minimal operator A in H = L 2 x l, x r )) is defined by Af)x) := d d fx) + vx)fx), 2 dx mx) dx f, dom A) := f H : m f W,2 x l, x r )) fx l ) = fx r ) = 0 m f ) x l ) = m f ), x r ) = 0 2.9) where it is assumed that the effective mass m satisfies m > 0 and m, m L x l, x r )), and that v L x l, x r )) is a real function. It is well known that A is a densely defined closed simple symmetric operator in H with deficiency indices n + A) = n A) = 2. The adjoint operator A is given by A f)x) = d d fx) + vx)fx), 2 dx mx) dx dom A ) = { f H : f, m f W,2 x l, x r )) }. It is straightforward to verify that Π A = {C 2, Γ 0, Γ }, where ) fxl ) Γ 0 f := and Γ fx r ) f := m f ) x l ) 2 m f ) x r ) f dom A ), is a boundary triplet for A. Note, that the selfadjoint extension A 0 := A kerγ 0 ) corresponds to Dirichlet boundary conditions, that is, dom A 0 ) = { f H : f, m f W,2 x l, x r )), fx l ) = fx r ) = 0 }. 2.0) The selfadjoint extension A corresponds to Neumann boundary conditions, i.e., { f, dom A ) = f H : m f } W,2 x l, x r )), m f )x l ) = m f. 2.) )x r ) = 0 Let ϕ λ and ψ λ, λ C, be the fundamental solutions of the homogeneous differential equation d d 2 dx m dxu + v u = λu satisfying the boundary conditions ϕ λ x l ) =, 2m ϕ λ)x l ) = 0 and ψ λ x l ) = 0, 2m ψ λ)x l ) =. Note that ϕ λ and ψ λ belong to L 2 x l, x r )) since x l, x r ) is a finite interval. A straightforward computation shows A0 λ) f ) x xr x) = ϕ λ x) ψ λ t)ft) dt + ψ λ x) ϕ λ t)ft) dt x l ϕ λx r ) ψ λ x r ) ψ λx) x xr x l ), ψ λ t)ft) dt
R-MATRIX 9 for x x l, x r ), f L 2 x l, x r )) and all λ ρa 0 ). In order to calculate the γ-field and Weyl function corresponding to Π A = {C 2, Γ 0, Γ } note that every element f λ N λ = kera λ) admits the representation f λ x) = ξ 0 ϕ λ x) + ξ ψ λ x), x x l, x r ), λ C, ξ 0, ξ C, where the coefficients ξ 0, ξ are uniquely determined. The relation ) ) 0 ξ0 Γ 0 f λ = ϕ λ x r ) ψ λ x r ) yields ψ λ x r ) ) ψλ x r ) 0 Γ ϕ λ x r ) 0 f λ = for ψ λ x r ) 0 that is λ σa 0 )) and it follows that the γ-field is given by ξ ξ0 γλ) : C 2 L 2 x l, x r )), ) ξ0 ) ϕλ )ψ ξ λ x r ) ψ λ )ϕ λ x r ))ξ 0 + ψ λ )ξ. ψ λ x r ) We remark that the adjoint operator admits the representation xr γλ) ϕλ x f = l y) ψ λ x r ) ψ λ y) ϕ λ x r ) ) ) fy) dy xr, ψ λ x r ) ψ λ y)fy) dy f L 2 x l, x r )). The Weyl function Mλ) = Γ γλ), λ ρa 0 ), then becomes ) ϕλ x Mλ) = r ) ψ λ x r ) 2m ψ λ )x. r) All selfadjoint extension of A can now be described with the help of selfadjoint relations Θ = Θ in C 2 via 2.4)-2.5) and their resolvents can be expressed in terms of the resolvent of A 0, the Weyl function M ) and the γ-field γ ), cf. Theorem 2.4. We leave the general case to the reader and note only that if Θ is a selfadjoint matrix of the form ) κl 0 Θ =, κ 0 κ l, κ r R, r then dom A Θ ) = x l { f dom A ) : ξ ) 2m f } )x l ) = κ l fx l ) 2m f )x r ) = κ r fx r ) and ) Θ Mλ) = κr ψ λ x r ) + 2m ψ λ )x ) r). ψ λ x r ) detθ Mλ)) κ l ψ λ x r ) + ϕ λ x r ) Obviously the case κ l = κ r = 0 leads to the Neumann operator A.
R-MATRIX 0 2.3.2 Infinite intervals Next we consider a singular problem on the infinite interval, x l ) in the Hilbert space K l = L 2, x l )). The minimal operator is defined by T l g l )x) := d d 2 dx m l x) dx g lx) + v l x)g l x), { g l, m dom T l ) := g l K l : l g l W,2, x l )) g l x l ) = m l g l) xl ) = 0 }, where m l > 0, m l, m l L, x l )) and v l L, x l )) is real. Then T l is a densely defined closed simple symmetric operator with deficiency indices n T l ) = n + T l ) =, see e.g. [39] and [8] for the fact that T l is simple. The adjoint operator T is given by Tl g l )x) = d d 2 dx m l x) dx g lx) + v l x)g l x), { } dom Tl ) = g l K l : g l, m l g l W,2, x l )). One easily verifies that Π Tl = {C, Υ l 0, Υ l }, ) Υ l 0g l := g l x l ) and Υ l g l := g l x l ), g l dom Tl ), 2m l is a boundary triplet for Tl. Let ϕ λ,l and ψ λ,l be the fundamental solutions of the equation d d 2 dx m l dx u + v lu = λu satisfying the boundary conditions ) ) ϕ λ,l x l ) =, x l ) = 0 and ψ λ,l x l ) = 0, x l ) =. 2m l ϕ λ,l 2m l ψ λ,l Then there exists a scalar function m l such that for each λ C \ R the function x g λ,l x) := ϕ λ,l x) m l λ)ψ λ,l x) belongs to L 2, x l )), cf. [39]. The function m l is usually called the Titchmarsh-Weyl function or Titchmarsh-Weyl coefficient and in our setting m l coincides with the Weyl function of the boundary triplet Π Tl = {C, Υ l 0, Υ l }, since Υ l g λ,l = m l λ)υ l 0g λ,l, g λ,l N λ,l := kert l λ), λ C \ R. An analogous example is the Schrödinger operator on the infinite interval x r, ) in K r = L 2 x r, )) defined by T r g r )x) := d d 2 dx m r x) dx g rx) + v r x)g r x), { g r, m dom T r ) := g r K r : r g r W,2 x r, )) g r x r ) = m r g r) xr ) = 0 },
R-MATRIX where m r > 0, m r, m r L x r, )) and v r L x r, )) is real. The adjoint operator Tr is and Π Tr = {C, Υ r 0, Υ r }, Tr g r )x) = d d 2 dx m r x) dx g rx) + v r x)g r x), { } dom Tr ) = g r K r : g r, m r g r W,2 x r, )) Υ r 0g r := g r x r ) and Υ r g r := ) g r x r ), g r dom Tr ), 2m r is a boundary triplet for Tr. Let ϕ λ,r and ψ λ,r be the fundamental solutions of the equation d d 2 dx m r dx u + v ru = λu satisfying the boundary conditions ) ) ϕ λ,r x r ) =, x r ) = 0 and ψ λ,r x r ) = 0, x r ) =. 2m r ϕ λ,r 2m r ψ λ,r Then there exists a scalar function m r such that for each λ C \ R the function x g λ,r x) := ϕ λ,r x) + m r λ)ψ λ,r x) belongs to L 2 x r, )). As above m r coincides with the Weyl function of the boundary triplet Π Tr := {C, Υ r 0, Υ r }. For our purposes it is useful to consider the direct sum of the two operators T l and T r. To this end we introduce the Hilbert space K := L 2, x l ) x r, )) =K l K r. An element g K will be written in the form g = g l g r, where g l L 2, x l )) and g r L 2 x r, )). The operator T = T l T r in K is defined by d d 2 dx m T g)x) = l x) dx g ) lx) + v l g l x) 0 0 d d 2 dx m rx) dx g, rx) + v r g r x) dom T ) = dom T l ) dom T r ), and T is a densely defined closed simple symmetric operator in K with deficiency indices n + T ) = n T ) = 2. The adjoint operator T is given by d d T 2 dx m g)x) = l x) dx g ) lx) + v l g l x) 0 0 d d 2 dx m rx) dx g, rx) + v r g r x) dom T ) = dom T l ) dom T r ). One easily checks that Π T = {C 2, Υ 0, Υ }, Υ 0 := Υ l 0, Υ r 0), Υ := Υ l, Υ r ), that is, ) gl x Υ 0 g = l ) and Υ g r x r ) g = ) m l g l ) xl ) 2 m r g r), xr )
R-MATRIX 2 g dom T ), is a boundary triplet for T. Note that T 0 = T kerυ 0 ) is the restriction of T to the domain dom T 0 ) = { g dom T ) : g l x l ) = g r x r ) = 0 }, that is, T 0 corresponds to Dirichlet boundary conditions at x l and x r. The Weyl function τ ) corresponding to the boundary triplet Π T = {C 2, Υ 0, Υ } is given by ) ml λ) 0 λ τλ) =, λ ρt 0 m r λ) 0 ). 3 Semibounded extensions and expansions in eigenfunctions Let A be a densely defined closed symmetric operator in the separable Hilbert space H and let {H, Γ 0, Γ } be a boundary triplet for A with γ-field γ ) and Weyl function M ). Fix some Θ = Θ CH) and let A Θ A be the corresponding selfadjoint extension via 2.4). In the next proposition it will be assumed that A 0 = A kerγ 0 ) and A Θ and hence also the symmetric operator A) are semi-bounded from below. Note that if A has finite defect it is sufficient for this to assume that A is semibounded, cf. Corollary 3.2. Proposition 3. Let A be a densely defined closed symmetric operator in H and let {H, Γ 0, Γ } be a boundary triplet for A with γ-field γ ) and Weyl function M ). Let A Θ be a selfadjoint extension of A corresponding to Θ = Θ CH) and assume that A 0 = A kerγ 0 ) and A Θ are semibounded from below. Then A Θ A 0 holds if and only if ran γλ) Θ Mλ) ) ) dom AΘ λ ) 3.) is satisfied for all λ < min{inf σa 0 ), inf σa Θ )}. Proof. Let A Θ A 0. From 2.8) we get A Θ λ) A 0 λ) = γλ) Θ Mλ) ) γλ) 0 for λ < min{inf σa 0 ), inf σa Θ )} which yields Θ Mλ) ) 0. By [6, Corollary 7-2] there is a contraction Y acting from H into H such that Θ Mλ) ) /2γλ) = Y A Θ λ) /2. Since λ R the adjoint has the form γλ) Θ Mλ) ) /2 = AΘ λ) /2 Y,
R-MATRIX 3 so that Therefore ran γλ) Θ Mλ) ) /2) dom AΘ λ ). ran γλ) Θ Mλ) ) ) ran γλ) Θ Mλ) ) /2) dom AΘ λ ) and 3.) is proved. Conversely, let us assume that condition 3.) is satisfied. λ < min{inf σa 0 ), inf σa Θ )} the operator FΘλ) := A Θ λ γλ) Θ Mλ) ) Then for each 3.2) is well defined on H and closed, and hence bounded. Besides FΘ λ) we introduce the densely defined operator F Θ λ) = Γ 0 A Θ λ) /2, dom F Θ λ)) = { f H : A Θ λ) /2 f dom A ) } 3.3) for λ < inf σa Θ ). It follows from 2.8), A 0 = A kerγ 0 ) and Γ 0 γλ) = I H that Γ 0 A Θ λ) = Θ Mλ) ) γλ) 3.4) holds for all λ ρa 0 ) ρa Θ ). Thus for λ < min{inf σa 0 ), inf σa Θ )} 3.2) becomes F Θλ) = A Θ λ Γ 0 A Θ λ) ) and together with 3.3) we conclude F Θ λ) = Γ 0 A Θ λ) /2 AΘ λ Γ 0 A Θ λ) ) ) = F Θ λ) ). This implies that F Θ λ) admits a bounded everywhere defined extension F Θ λ) for λ < min{inf σa 0 ), inf σa Θ )} such that F Θ λ) = F Θ λ) = FΘ λ). From 3.4) and Mλ) = Mλ) we find Γ 0 Γ0 A Θ λ) ) = Θ Mλ) ), λ ρa0 ) ρa Θ ), so that for λ < min{inf σa 0 ), inf σa Θ )} Θ Mλ) ) = Γ0 A Θ λ) /2 A Θ λ Γ 0 A Θ λ) ) = F Θ λ)f Θ λ) 0. Using 2.8) we find A Θ λ) A 0 λ) for λ < min{inf σa 0 ), inf σa Θ )} which yields A Θ A 0.
R-MATRIX 4 Corollary 3.2 Let A be a densely defined closed symmetric operator in H and let {H, Γ 0, Γ } be a boundary triplet for A with γ-field γ ) and Weyl function M ). Assume that A has finite defect and that A 0 = A kerγ 0 ) is the Friedrichs extension. Then every selfadjoint extension A Θ of A in H is semibounded from below and ran γλ) Θ Mλ) ) ) dom AΘ λ) is satisfied for all λ < min{inf σa 0 ), inf σa Θ )}. In the next proposition we obtain a representation of the function λ Θ Mλ)) in terms of eigenvalues and eigenfunctions of A Θ. This representation will play an important role in Section 5. Proposition 3.3 Let A be a densely defined closed symmetric operator in H and let {H, Γ 0, Γ } be a boundary triplet for A with Weyl function M ). Let A Θ be a selfadjoint extension of A corresponding to Θ = Θ CH) and assume that A 0 = A kerγ 0 ) and A Θ are semibounded from below, A Θ A 0, and that the spectrum of A Θ is discrete. Then the [H]-valued function λ Θ Mλ)) admits the representation ) Θ Mλ) = λ k λ), Γ 0 ψ k )Γ 0 ψ k, λ ρa 0 ) ρa Θ ), 3.5) where {λ k }, k =, 2,..., are the eigenvalues of A Θ in increasing order and {ψ k } are the corresponding eigenfunctions. The convergence in 3.5) is understood in the strong sense. Proof. Let λ 0 < min{inf σa 0 ), inf σa Θ )} and let E m, m N, be the orthogonal projection in H onto the subspace spanned by the eigenfunctions {ψ k }, k =,..., m <, of A Θ. Considerations similar as in the proof of Proposition 3. show Γ 0 E m γλ 0 ) Θ Mλ 0 ) ) = Γ 0 A Θ λ 0 ) /2 E m AΘ λ 0 γλ 0 ) Θ Mλ 0 ) ) = F Θ λ 0 )E m F Θ λ 0 ), where F Θ λ 0 ) is defined as in 3.3) and F Θ λ 0 ) [H, H] denotes the closure. Hence we have lim Γ 0E m γλ 0 ) Θ Mλ 0 ) ) = F Θ λ 0 )F Θ λ 0 ) = Θ Mλ 0 ) ) m in the strong topology. For λ ρa 0 ) ρa Θ ) we conclude from the representations Θ Mλ) ) = Γ0 Γ0 A Θ λ) ) = F Θ λ 0 )A Θ λ 0 )A Θ λ) F Θ λ 0 )
R-MATRIX 5 and that Γ 0 E m γλ) Θ Mλ) ) = F Θ λ 0 )A Θ λ 0 )A Θ λ) E m F Θ λ 0 ) lim Γ 0E m γλ) Θ Mλ) ) ) = Θ Mλ) m in the strong sense for all λ ρa 0 ) ρa Θ ). Further, since the resolvent of A Θ admits the representation A Θ λ) = λ k λ), ψ k )ψ k, λ ρa Θ ), where the convergence is in the strong sense, we find Γ 0 A Θ λ) E m = m λ k λ), ψ k )Γ 0 ψ k. For λ ρa 0 ) ρa Θ ) the adjoint operator is given by E m Γ0 A Θ λ) ) ) γλ) = Em Θ Mλ) ) = Em γλ) Θ Mλ) ) m = λ k λ), Γ 0 ψ k )ψ k. Here we have again used 2.8), A 0 = A kerγ 0 ) and Γ 0 γλ) = I H. Replacing λ by λ and applying Γ 0 we obtain from the above formula the representation Γ 0 E m γλ) Θ Mλ) ) m = λ k λ), Γ 0 ψ k )Γ 0 ψ k for all λ ρa 0 ) ρa Θ ). By the above arguments the left hand side converges in the strong sense to Θ Mλ)). Therefore we obtain 3.5). The special case Θ = 0 [H] will be of particular interest in our further investigations. In this situation Proposition 3.3 reads as follows. Corollary 3.4 Let A be a densely defined closed symmetric operator in H and let {H, Γ 0, Γ } be a boundary triplet for A with Weyl function M ). Assume that A 0 = A kerγ 0 ) and A = A kerγ ) are semibounded from below, A A 0, and that σa ) is discrete. Then the [H]-valued function λ Mλ) admits the representation Mλ) = λ λ k ), Γ 0 ψ k )Γ 0 ψ k, λ ρa 0 ) ρa ), 3.6) where {λ k }, k =, 2,..., are the eigenvalues of A in increasing order, {ψ k } are the corresponding eigenfunctions, and the convergence in 3.6) is understood in the strong sense.
R-MATRIX 6 Proposition 3.3 and Corollary 3.4 might suggest that the Weyl function M can be represented as a convergent series involving the eigenvalues and eigenfunctions of the selfadjoint operator A 0. The following proposition shows that this is not possible if A 0 is chosen to be the Friedrichs extension. Proposition 3.5 Let A be a densely defined closed symmetric operator in H with finite or infinite deficiency indices and let {H, Γ 0, Γ } be a boundary triplet for A. Assume that A 0 = A kerγ 0 ) and A = A kerγ ) are semibounded, that A 0 coincides with the Friedrichs extension of A and that σa 0 ) is discrete. Then the limit lim m m λ µ k ), Γ φ k )Γ φ k, λ ρa 0 ), where {µ k }, k =, 2,..., are the eigenvalues of A 0 in increasing order and {φ k } are the corresponding eigenfunctions, does not exist. Proof. We set and Qλ) := Γ A 0 λ), λ ρa 0 ), 3.7) Gλ) := Γ Qλ) = Γ Γ A 0 λ) ), λ ρa0 ). Taking into account the relation A λ) = A 0 λ) γλ)mλ) γλ), λ ρa 0 ) ρa ), and 2.6) we find Qλ) = γλ) and Gλ) = Mλ) λ ρa 0 ) ρa ). Let m N, let E m be the projection onto the subspace spanned by the eigenfunctions {φ k }, k =,..., m, and define Q m λ) := Qλ)E m and G m λ) := Γ E m Qλ), λ ρa 0 ). With the help of A 0 λ) = and 3.7) we find the representation G m λ) = µ k λ), φ k )φ k m µ k λ), Γ φ k ) Γ φ k, λ ρa 0 ) ρa ), and on the other hand G m λ) = Q m λ)a 0 λ)e m Qλ) = γλ) A 0 λ)e m γλ)
R-MATRIX 7 for λ ρa 0 ) ρa ). Let λ R, λ < min{inf σa 0 ), inf σa )}, and assume that there is an element η H such that the limit lim m Gm λ)η = lim m exists. Since for h := γλ)η N λ = kera λ) m µ k λ) η, Γ φ k ) Γ φ k 3.8) G m λ)η, η) = A 0 λ)e m γλ)η, γλ)η ) = A 0 λe m h we obtain from 3.8) that the limit lim m A 0 λe m h exists and is finite. Therefore there is a subsequence {m n }, n N, such that g := w- lim A0 λ E mn h and lim E m n h = h. n n Hence we conclude h dom A 0 λ) and g = A 0 λ h. But according to [, Lemma 2.] we have dom A 0 λ) N λ = {0}, so that h = 0 and therefore η = 0. 2 4 Scattering theory and representation of S and R-matrices Let A be a densely defined closed simple symmetric operator in the separable Hilbert space H and assume that the deficiency indices of A coincide and are finite, n + A) = n A) <. Let {H, Γ 0, Γ } be a boundary triplet for A, A 0 = A kerγ 0 ), and let A Θ be a selfadjoint extension of A which corresponds to a selfadjoint relation Θ CH). Note that dim H = n ± A) is finite. Let P op be the orthogonal projection in H onto the subspace H op := dom Θ) and decompose Θ as in 2.2), Θ = Θ op Θ with respect to H op H. The Weyl function M ) corresponding to {H, Γ 0, Γ } is a matrix-valued Nevanlinna function and the same holds for N Θ λ) := Θ Mλ) ) = Θop M op λ) ) Pop, λ C\R, 4.) where M op λ) = P op Mλ)P op, cf. [25, page 37]. We will in general not distinguish between the orthogonal projection onto H op and the canonical embedding of H op into H. By Fatous theorem see [4, 7]) the limits Mλ + i0) := lim Mλ + iɛ) ɛ +0 and ) N Θ λ + i0) := lim Θ Mλ + iɛ) ɛ +0
R-MATRIX 8 from the upper half-plane exist for a.e. λ R. We denote the set of real points where the limits exist by Σ M and Σ NΘ, respectively, and we agree to use a similar notation for arbitrary scalar and matrix-valued Nevanlinna functions. It is not difficult to see that N Θ λ + i0) = Θ Mλ + i0) ) = Θop M op λ + i0) ) Pop, holds for all λ Σ M Σ NΘ and that R\Σ M Σ NΘ ) has Lebesgue measure zero, cf. [3, 2.3]. Since dim H is finite by 2.8) dim ran A Θ λ) A 0 λ) )) <, λ ρa Θ ) ρa 0 ), and therefore the pair {A Θ, A 0 } performs a so-called complete scattering system, that is, the wave operators W ± A Θ, A 0 ) := s- lim t ± eitaθ e ita0 P ac A 0 ), exist and their ranges coincide with the absolutely continuous subspace H ac A Θ ) of A Θ, cf. [2, 22, 39, 46]. P ac A 0 ) denotes the orthogonal projection onto the absolutely continuous subspace H ac A 0 ) of A 0. The scattering operator S Θ of the scattering system {A Θ, A 0 } is then defined by S Θ := W + A Θ, A 0 ) W A Θ, A 0 ). If we regard the scattering operator as an operator in H ac A 0 ), then S Θ unitary, commutes with the absolutely continuous part is A ac 0 := A 0 dom A 0 ) H ac A 0 ) of A 0 and it follows that S Θ is unitarily equivalent to a multiplication operator induced by a family {S Θ λ)} of unitary operators in a spectral representation of A ac 0, see e.g. [2, Proposition 9.57]. This family is called the scattering matrix of the scattering system {A Θ, A 0 }. In [4] a representation theorem for the scattering matrix {S Θ λ)} in terms of the Weyl function M ) was proved, which is of similar type as Theorem 4. below. We will make use of the notation H Mλ) := ran Im Mλ)) ), λ Σ M, 4.2) and we will usually regard H Mλ) as a subspace of H. The orthogonal projection onto H Mλ) will be denoted by P Mλ). Note that for λ ρa 0 ) R the Hilbert space H Mλ) is trivial by 2.7). The family {P Mλ) } λ Σ M of orthogonal projections in H onto H Mλ), λ Σ M, is measurable and defines an orthogonal projection in the Hilbert space L 2 R, dλ, H). The range of this projection is denoted by L 2 R, dλ, H Mλ) ). Let P op and M op λ) = P op Mλ)P op, λ Σ M, be as above. For each λ Σ M the space H Mλ) will also be written as the orthogonal sum of H Mopλ) = ran Im M op λ)) )
R-MATRIX 9 and H M opλ) := H Mλ) H Mopλ) = ker Im M op λ)) ). The following theorem is a variant of [4, Theorem 3.8]. The essential advantage here is, that the particular form of the scattering matrix {S Θ λ)} immediately shows that the multivalued part of the selfadjoint parameter Θ has no influence on the scattering matrix. Theorem 4. Let A be a densely defined closed simple symmetric operator with equal finite deficiency indices in the separable Hilbert space H and let Π = {H, Γ 0, Γ } be a boundary triplet for A with corresponding Weyl function M ). Furthermore, let A 0 = A kerγ 0 ) and let A Θ be a selfadjoint extension of A which corresponds to Θ = Θ op Θ CH) via 2.4). Then the following holds. i) The absolutely continuous part A ac 0 of A 0 is unitarily equivalent to the multiplication operator with the free variable in L 2 R, dλ, H Mλ) ). ii) With respect to the decomposition H Mλ) = H Mopλ) HM opλ) the scattering matrix {S Θ λ)} of the complete scattering system {A Θ, A 0 } in L 2 R, dλ, H Mλ) ) is given by S Θ λ) = SΘopλ) 0 0 I H Mopλ) ) [ H Mopλ) H M opλ)], where S Θop λ) = I HMopλ) + 2i Im M op λ)) Θ op M op λ) ) Im M op λ)) and λ Σ M Σ NΘ, M op λ) := M op λ + i0). Proof. Assertion i) was proved in [4, Theorem 3.8] and moreover it was shown that the scattering matrix { S Θ λ)} of the complete scattering system {A Θ, A 0 } in L 2 R, dλ, H Mλ) ) has the form S Θ λ) = I HMλ) + 2i Im Mλ)) Θ Mλ) ) Im Mλ)) [H Mλ) ] for all λ Σ M Σ NΘ. With the help of 4.) this becomes S Θ λ) = I HMλ) + 2i Im Mλ))P op Θop M op λ) ) Pop Im Mλ)). From the polar decomposition of Im Mλ))P op, λ Σ M, we obtain a family of isometric mappings V λ), λ Σ M, from H Mopλ) onto ran Im Mλ))P op ) defined by V λ) Im M op λ)) x := Im Mλ)) P op x
R-MATRIX 20 and we extend V λ) to a family Ṽ λ) of unitary mappings in H Mλ). Note that Ṽ λ) maps ker Im M op λ))) isometrically onto kerp op Im Mλ))). It is not difficult to see that the scattering matrix S Θ λ) := Ṽ λ) SΘ λ)ṽ λ), λ ΣM Σ NΘ, with respect to the decomposition H Mλ) = H Mopλ) HM opλ) as in assertion ii). is of the form We point out that the scattering matrix {S Θ λ)} of the complete scattering system {A Θ, A 0 } is defined for a.e. λ R and that in Theorem 4.ii) a special representative of the corresponding equivalence class was chosen. We also note that the operator Im M op λ)) is regarded as an operator in H Mopλ). Next we introduce the R-matrix {R Θ λ)} of the scattering system {A Θ, A 0 } in accordance with Blatt and Weiskopf [5], R Θ λ) := i I HMλ) S Θ λ) ) I HMλ) + S Θ λ) ) 4.3) for all λ Σ M Σ NΘ satisfying ρs Θ λ)). Since S Θ λ) is unitary it follows that R Θ λ) is a selfadjoint matrix. Note also that S Θ λ) = ii HMλ) R Θ λ) ) ii HMλ) + R Θ λ) ) 4.4) holds for all real λ where R Θ λ) is defined. The next theorem is of similar flavor as Theorem 4.. We express the R- matrix of the scattering system {A Θ, A 0 } in terms of the Weyl function M ) and the selfadjoint parameter Θ CH). Again we make use of the special space decomposition which shows that the pure relation part Θ has no influence on the R-matrix. Theorem 4.2 Let A be a densely defined closed simple symmetric operator with equal finite deficiency indices in the separable Hilbert space H and let Π = {H, Γ 0, Γ } be a boundary triplet for A with corresponding Weyl function M ). Furthermore, let A 0 = A kerγ 0 ) and let A Θ be a selfadjoint extension of A corresponding to Θ CH). Then for all λ Σ M Σ NΘ with the R-matrix of {A Θ, A 0 } is given by R Θ λ) = ker Θ op Re M op λ)) ) = {0} Im Mop λ)) Θ op Re M op λ)) ) ) Im Mop λ)) 0, 0 0 with respect to H Mλ) = H Mopλ) H M opλ), where M opλ) = M op λ + i0). Proof. It follows immediately from the definition 4.3) and the representation of the scattering matrix in Theorem 4. ii), that the R-matrix of {A Θ, A 0 }
R-MATRIX 2 is a diagonal block matrix with respect to the space decomposition H Mλ) = H Mopλ) HM and that the restriction of R opλ) Θλ) to HM opλ) is identically equal to zero. Moreover, for every λ Σ M Σ NΘ it follows from the representation of the scattering matrix that Im M op λ)) I HMopλ) + S Θop λ) ) = 2 { I HMopλ) + iim M op λ)) Θ op M op λ) ) } Im M op λ)) = 2 Θ op Re M op λ)) ) Θ op M op λ) ) Im M op λ)) holds. If λ Σ M Σ NΘ is such that Θ op Re M op λ)) is invertible, then we obtain Im M op λ)) I HMopλ) + S Θop λ) ) so that = 2 Θop M op λ) ) Θ op Re M op λ)) ) Im M op λ)), 2i Θ op M op λ) ) Im M op λ)) I HMopλ) + S Θop λ) ) = i Θ op Re M op λ)) ) Im M op λ)). Finally multiplication by Im M op λ)) from the left gives IHMopλ) S Θ op λ)) I HMopλ) + S Θ opλ) ) = i Im M op λ)) Θ op Re M op λ)) ) Im M op λ)) so that assertion i) follows immediately from the definition of the R-matrix in 4.3). 5 Scattering in coupled systems Let H and K be separable Hilbert spaces and let A and T be densely defined closed simple symmetric operators in H and K, respectively. We assume that the deficiency indices of A and T coincide and are finite, n := n + A) = n A) = n + T ) = n T ) <. Then there exist boundary triplets {H, Γ 0, Γ } and {H, Υ 0, Υ } for the adjoint operators A and T, respectively, with fixed selfadjoint extensions A 0 := A kerγ 0 ) and T 0 := T kerυ 0 ) 5.)
R-MATRIX 22 in H and K, respectively, and dim H = n. The Weyl functions of {H, Γ 0, Γ } and {H, Υ 0, Υ } will be denoted by M ) and τ ), respectively. Besides the spaces H Mλ), λ Σ M, see 4.2)) we will make use of the finite dimensional spaces and H τλ) = ran Im τλ + i0)) ), λ Σ τ, H M+τ)λ) = ran Im M + τ)λ + i0) )), λ Σ M+τ Σ M Σ τ ). In the following theorem we calculate the S and R-matrix of a special scattering system { L, L 0 } in H K in terms of the Weyl functions M and τ. Theorem 5. is in principle a consequence of Theorem 4. and Theorem 4.2, cf. [3, Theorem 4.5]. We note that the coupling procedure in the first part of the theorem is similar to the one in [9]. Theorem 5. Let A, {H, Γ 0, Γ }, M ) and T, {H, Υ 0, Υ }, τ ) be as above. Then the following holds. i) The pair { L, L 0 }, where L 0 := A 0 T 0 and { L = A T f g dom A T ) : Γ 0 f Υ 0 g = 0 Γ f + Υ g = 0 }, 5.2) forms a complete scattering system in the Hilbert space H K and L ac 0 is unitarily equivalent to the multiplication operator with the free variable in L 2 R, dλ, H Mλ) H τλ) ). ii) With respect to the decomposition H M+τ)λ) H M+τ)λ) 5.3) of H Mλ) H τλ) the scattering matrix { Sλ)} of { L, L 0 } is given by ) Sλ) 0 Sλ) = 0 I [ H H M+τ)λ) HM+τ)λ)], M+τ)λ) where Sλ) = I HM+τ)λ) 2i Im Mλ) + τλ)) Mλ) + τλ) ) Im Mλ) + τλ)) and λ Σ M Σ τ Σ M+τ), Mλ) := Mλ + i0), τλ) = τλ + i0). iii) For all λ Σ M Σ τ Σ M+τ) R-matrix of { L, L 0 } is given by with kerre Mλ) + τλ))) = {0} the Rλ) = p p «Im Mλ) + τλ))`re Mλ) + τλ)) Im Mλ) + τλ)) 0 0 0 with respect to the decomposition 5.3).
R-MATRIX 23 Proof. i) Let L := A T, so that L is a densely defined closed simple symmetric operator in the Hilbert space H K. Clearly, L has deficiency indices n ± L) = 2n, and it is easy to see that { H, Γ 0, Γ }, where ) Γ0 f Γ 0 f g) :=, Γ f g) := Υ 0 g Γ f Υ g ) and H := H H, f dom A ), g dom T ), is a boundary triplet for the adjoint operator L = A T in H K. Together with the selfadjoint operators A 0 and T 0 from 5.) we obviously have It is not difficult to verify that { ) x, x) Θ := y, y) L 0 := L ker Γ 0 ) = A 0 T 0. } : x, y H C H H ) 5.4) is a selfadjoint relation in H and that the corresponding selfadjoint extension L Γ ) Θ in H K via 2.4) coincides with the operator L in 5.2), cf. [3]. Since L has finite deficiency indices, L is finite rank perturbation of L 0 in resolvent sense cf. Theorem 2.4 and Section 4), and hence { L, L 0 } is a complete scattering system in H K. Moreover, as the Weyl function M ) of { H, Γ 0, Γ } is given by ) Mλ) 0 Mλ) =, λ ρl 0 τλ) 0 ), 5.5) it follows from Theorem 4. i) that the absolutely continuous part L ac 0 of L 0 is unitarily equivalent to the multiplication operator with the free variable in L 2 R, dλ, H fmλ) ) = L 2 R, dλ, H Mλ) H τλ) ). ii)-iii) Note that the operator part Θ op of the selfadjoint relation Θ in 5.4) is defined on H op := dom Θ) = { x, x) : x H } and that Θ op = 0 [ H op ], cf. 2.2). Next we will calculate the [ H op ]-valued function M op ), and in order to avoid possible confusion we will distinguish between embeddings and projections here. The canonical embedding of Hop into H H is given by ι op : H op H H, y ) y, 2 y and the adjoint ι op [H H, H op ] is the orthogonal projection P op from H H onto H op, P op u v) = 2 u + v). Then we obtain M op λ) = P op Mλ)ιop = 2 Mλ) + τλ) ), λ ρl0 ),
R-MATRIX 24 from 5.5). Now the assertions ii) and iii) follow easily from Theorem 4. ii) and Theorem 4.2, respectively. The case that the operator A 0 has discrete spectrum is of particular importance in several applications. In this situation Theorem 5. reduces to the following corollary. Corollary 5.2 Let the assumptions and { L, L 0 } be as in Theorem 5. and assume, in addition, that σa 0 ) is discrete. Then the following holds. i) L ac 0 is unitarily equivalent to the multiplication operator with the free variable in L 2 R, dλ, H τλ) ) ii) The scattering matrix {Sλ)} of { L, L 0 } in L 2 R, dλ, H τλ) ) is given by Sλ) = I Hτλ) 2i Im τλ)) Mλ) + τλ) ) Im τλ)) for λ Σ M Σ τ Σ M+τ), where Mλ) := Mλ + i0), τλ) = τλ + i0). iii) For all λ Σ M Σ τ Σ M+τ) R-matrix of { L, L 0 } is given by with kermλ) + Re τλ))) = {0} the Rλ) = Im τλ)) Mλ) + Re τλ)) ) Im τλ)) Proof. The assumption σa 0 ) = σ p A 0 ) implies Im Mλ)) = {0} for all λ Σ M. Therefore H M+τ)λ) = H τλ) and H Mλ) = {0}, λ Σ M, and the statements follow immediately from Theorem 5.. From relation 4.4) we obtain the next corollary. We note that this statement can be formulated also for the case when σa 0 ) is not discrete. However in our applications we will only make use of the more special variant below. Corollary 5.3 Let the assumptions be as in Corollary 5.2. Then for all λ Σ M Σ τ Σ M+τ) with kermλ) + Re τλ))) = {0} the scattering matrix {Sλ)} of { L, L 0 } admits the representation Sλ) = ii Hτλ) + Im τλ)) Mλ) + Re τλ)) ) ) Im τλ)) ii Hτλ) Im τλ)) Mλ) + Re τλ)) ) ) Im τλ)) and, if, in particular, Re τλ)) = 0, then Sλ) = ii Hτλ) + Im τλ))mλ) ) Im τλ)) ii Hτλ) Im τλ))mλ). Im τλ)))
R-MATRIX 25 Our next objective is to express the scattering matrix of the scattering system { L, L 0 } in terms of the eigenfunctions of a family of selfadjoint extensions of A. For this let again τ ) be the Weyl function of {H, Υ 0, Υ }, let µ Σ τ, and let {H, Γ 0, Γ } be a boundary triplet for A as in the beginning of this section. Then Re τµ)) is a selfadjoint matrix in H and therefore the operator A Re τµ)) = A ker ) Γ + Re τµ))γ 0 5.6) is a selfadjoint extension of A in H, cf. Proposition 2.2. Note that by Theorem 2.4 a point λ ρa 0 ) belongs to ρa Re τµ)) ) if and only if 0 ρmλ) + Re τµ)) holds. The following corollary is a reformulation of Proposition 3.3 in our particular situation. Corollary 5.4 Let A, {H, Γ 0, Γ }, M ) and T, {H, Υ 0, Υ }, τ ) be as above and assume σa 0 ) = σ p A 0 ) and that A is semibounded from below. For each µ Σ τ with A Re τµ)) A 0 the function λ Re τµ)) + Mλ)) admits the representation Mλ) + Re τµ)) ) = λ k [µ] λ), Γ 0 ψ k [µ] ) Γ 0 ψ k [µ], where {λ k [µ]}, k =, 2,..., are the eigenvalues of the selfadjoint extension A Re τµ)) in increasing order and ψ k [µ] are the corresponding eigenfunctions. Setting µ = λ in Corollary 5.4 and taking into account Corollary 5.2 and Corollary 5.3 we obtain the following representations of the R-matrix and scattering matrix of { L, L 0 }. Theorem 5.5 Let the assumptions be as in Corollary 5.4. Then for all λ Σ M Σ τ Σ M+τ) with kermλ) + Re τλ))) = {0} and A Re τλ)) A 0 the R-matrix and the scattering matrix of { L, L 0 } admit the representations and Rλ) = λ k [λ] λ) Im τλ)), Γ 0 ψ k [λ] ) Im τλ))γ 0 ψ k [λ] Sλ) = ii Hτλ) ii Hτλ) + λ k [λ] λ) Im τλ)), Γ 0 ψ k [λ] ) ) Im τλ))γ 0 ψ k [λ] λ k [λ] λ) Im τλ)), Γ 0 ψ k [λ] ), Im τλ))γ 0 ψ k [λ]) respectively, where {λ k [λ]}, k =, 2,..., are the eigenvalues of the selfadjoint extension A Re τλ)) in increasing order and ψ k [λ] are the corresponding eigenfunctions.
R-MATRIX 26 If Re τλ)) = 0 for some λ Σ τ, then the operator A Re τλ)) in 5.6) coincides with the selfadjoint operator A = A kerγ ). This yields the next corollary. Corollary 5.6 Let the assumptions be as in Corollary 5.4. Then for all λ Σ M Σ τ Σ M+τ) with Re τλ)) = 0, kermλ)) = {0} and A A 0 the R-matrix and the scattering matrix of { L, L 0 } admit the representations and Rλ) = λ k λ) ) Im τλ)), Γ 0 ψ k Im τλ))γ0 ψ k Sλ) = ii Hτλ) ii Hτλ) + λ k λ) ) ) Im τλ)), Γ 0 ψ k Im τλ))γ0 ψ k λ k λ) ) ), Im τλ)), Γ 0 ψ k Im τλ))γ0 ψ k respectively, where {λ k }, k =, 2,..., are the eigenvalues of the selfadjoint extension A in increasing order and ψ k are the corresponding eigenfunctions. Remark 5.7 The assumption A A 0 in Corollary 5.6 above is necessary. Indeed, let us assume that A 0 A and that A is the Friedrichs extension. Let us show that in this case the sum λ k λ), Γ 0 ψ k )Γ 0 ψ k 5.7) cannot converge, where {λ k } and {ψ k } are the eigenvalues and eigenfunctions of A. For this consider the boundary triplet {H, Γ 0, Γ }, Γ 0 = Γ and Γ = Γ 0. Obviously A 0 = A kerγ 0) = A, A = A kerγ ) = A 0 and A 0 is the Friedrichs extension. By Proposition 3.5 we obtain that the sum λ λ k ), Γ ψ k )Γ ψ k diverges, where {λ k } and {ψ k } are the eigenvalues and eigenfunctions of A 0 = A. Using Γ = Γ 0 one gets that the sum 5.7) diverges. 6 Scattering systems of differential operators In this section we illustrate the general results from the previous sections for scattering systems which consist of regular and singular second order differential operators, see Section 2.3.
R-MATRIX 27 6. Coupling of differential operators Let the symmetric operators A = 2 T = T l T r = d 2 dx m l d d dx m d dx + v l dx + v and ) d 2 dx m r ) d dx + v r in H = L 2 x l, x r )) and K = L 2, x l )) L 2 x r, )) and the boundary triplets Π A = {C 2, Γ 0, Γ } and Π T = {C 2, Υ 0, Υ } be as in Section 2.3. and and Section 2.3.2, respectively. By Theorem 5.i) the operator { } L := A T f g dom A T Γ ) : 0 f Υ 0 g = 0 6.) Γ f + Υ g = 0 is a selfadjoint extension of L = A T in H K. We can identify H K with L 2 x l, x r )) L 2, x l )) L 2 x r, )) = L 2 R). The elements f g in H K, f H, g = g l g r K, will be written as f g l g r. Here the conditions Γ 0 f = Υ 0 g and Γ f = Υ g, f dom A ), g dom T ), explicitely mean g l x l ) = fx l ) and fx r ) = g r x r ), and ) ) m f x l ) = g l x l ) and m l ) ) m f x r ) = g r x r ). m r Hence the selfadjoint operator 6.) has the form Lf g l g r ) = d d 2 dx mx) dx f + vf 0 0 0 d d 2 dx m l dx g l + v l g l 0 0 0 d 2 dx and coincides with the usual Schrödinger operator { f L 2 R) : f, d d 2 dx m dx + ṽ m r m f W,2 R) d dx g r + v r g r }, where and mx), x x l, x r ) mx) := m l x), x, x l ) m r x), x x r, ) vx), x x l, x r ) ṽx) := v l x), x, x l ) v r x), x x r, ).