Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps
aa r X i v : . [ m a t h . SP ] M a y Jordan chains of elliptic partial differential operatorsand Dirichlet-to-Neumann maps
J. Behrndt and A.F.M. ter Elst Abstract
Let Ω ⊂ R d be a bounded open set with Lipschitz boundary Γ. It willbe shown that the Jordan chains of m-sectorial second-order ellipticpartial differential operators with measurable coefficients and (localor non-local) Robin boundary conditions in L (Ω) can be character-ized with the help of Jordan chains of the Dirichlet-to-Neumann mapand the boundary operator from H / (Γ) into H − / (Γ). This resultextends the Birman–Schwinger principle in the framework of ellipticoperators for the characterization of eigenvalues, eigenfunctions andgeometric eigenspaces to the complete set of all generalized eigenfunc-tions and algebraic eigenspaces.May 20192010 Mathematics Subject Classification: 35J57, 35P05, 47A75, 47F05.Keywords: Jordan chain, eigenvector, generalized eigenvector, Robin boundary condition,Dirichlet-to-Neumann operator Home institutions:
1. Institut f¨ur Angewandte Mathematik 2. Department of MathematicsTechnische Universit¨at Graz University of AucklandSteyrergasse 30 Private bag 92019A-8010 Graz Auckland 1142Austria New ZealandEmail: [email protected] Email: [email protected]
Introduction
The Dirichlet-to-Neumann map is an important object in the analysis of elliptic partialdifferential equations since it can be used to describe the spectra of the associated ellipticoperators. The principal strategy and advantage is that a spectral problem for a partialdifferential operator on a domain Ω is reduced to a spectral problem for an operatorfunction on the boundary Γ of this domain, where, very roughly speaking, the Dirichletand Neumann data can be measured . This type of approach to problems in spectral andscattering theory for elliptic partial differential operators was used in the self-adjoint casein, e.g. [AM, BMN, BR1, BR2, GM1, GM3, GMZ, MPP, Marl, MPPRY, MPP, Post],for non-self-adjoint situations in, e.g. [BGHN, BGW, Gru, Mal], and we also refer thereader to the more abstract contributions [AE2, AE4, AE5, AEKS, AEW, BMN, BHMNW,BMNW1, BGP, DHK, DM1, DM2, EO1, EO2, LT, MM, Posi].In the present paper we are interested in a characterization of Jordan chains of eigen-values of elliptic operators. To motivate our investigations let us consider here in theintroduction only the special case of a Schr¨odinger operator A = − ∆ + V on a boundedLipschitz domain Ω ⊂ R d with d ≥ V ∈ L ∞ (Ω).Later in this paper much more general second-order partial differential expressions A withmeasurable coefficients will be considered; see Section 3 for details. The Dirichlet-to-Neumann map D ( λ ) corresponding to − ∆ + V can be defined as a bounded operator D ( λ ) : H / (Γ) → H − / (Γ) by Tr f λ γ N f λ , where f λ ∈ H (Ω) is such that A f λ = λf λ . Here Tr f λ ∈ H / (Γ) and γ N f λ ∈ H − / (Γ)denote the Dirichlet and Neumann trace of f λ , respectively, and λ ∈ C is not an eigenvalueof the Dirichlet realization A D of − ∆ + V . Assume for simplicity that B : L (Γ) → L (Γ)is a bounded operator and consider the (non-local) Robin realization of − ∆ + V definedby A B f = − ∆ f + V f, dom A B = (cid:8) f ∈ H (Ω) : γ N f = B Tr f and − ∆ f + V f ∈ L (Ω) (cid:9) . (1.1)Note that the resolvents of A D and A B are both compact operators in L (Ω) due to thecompactness of the embedding H (Ω) ֒ → L (Ω) and hence the spectra of A D and A B arediscrete. It is well-known and easy to see that for all λ σ p ( A D ) one has λ ∈ σ p ( A B )if and only if ker ( D ( λ ) − B ) = { } . Sometimes this is referred to as a variant of theBirman–Schwinger principle. In fact, if λ ∈ σ p ( A B ) and f ∈ dom A B is a correspondingeigenfunction, then Tr f = 0 (as otherwise f would be an eigenfunction for A D at λ ) and( D ( λ ) − B )Tr f = D ( λ )Tr f − B Tr f = γ N f − B Tr f = 0 , and conversely, if ϕ ∈ ker ( D ( λ ) − B ) \ { } , then the unique solution f ∈ H (Ω) of theboundary value problem ( − ∆ + V ) f = λ f with Tr f = ϕ , satisfies γ N f − B Tr f = 0,so that f ∈ dom A B is an eigenfunction of A B corresponding to λ .2n the situation where the potential V is not real-valued or the Robin boundary op-erator B is not symmetric the Schr¨odinger operator A B in (1.1) is m-sectorial, but notself-adjoint in L (Ω). Therefore, in general, the eigenvalues of A B are not semisimpleand besides an eigenvector f also (finitely many) generalized eigenvectors f , . . . , f k areassociated to an eigenvalue λ , which form a so-called Jordan chain. It is the main ob-jective of the present paper to analyse the Jordan chains f , f , . . . , f k corresponding toan eigenvalue λ of A B with the help of the Dirichlet-to-Neumann operator in a sim-ilar form as in the above mentioned Birman–Schwinger principle. In fact, using thenotion of Jordan chains for holomorphic operator functions due to M.V. Keldysh [Kel](see also [Mark, § { f , f , . . . , f k } form a Jordan chain of A B at λ ∈ σ p ( A B ) ∩ ρ ( A D ) if and only if the correspondingtraces ϕ = Tr f , ϕ = Tr f , . . . , ϕ k = Tr f k form a Jordan chain for the holomorphic L ( H / (Γ) , H − / (Γ))-valued operator function λ M ( λ ) = D ( λ ) − B at λ , that is, j X l =0 l ! M ( l ) ( λ ) ϕ j − l = 0 (1.2)for all j ∈ { , . . . , k } , where M ( l ) ( λ ) denotes the l -th derivative of the function M at λ .Note that for j = 0 the characterization of the eigenvector f in the Birman–Schwingerprinciple follows from (1.2); see the above discussion or Corollary 4.2.The structure of this paper is as follows. In Section 2 we briefly recall the notion ofJordan chains for operators and holomorphic operator functions. In Section 3 we introducethe elliptic differential operators and the corresponding Dirichlet-to-Neumann map that isused for the analysis of the algebraic eigenspaces. Here we treat second-order divergenceform elliptic operators with (complex) L ∞ -coefficients of the form A = − d X k,l =1 ∂ k c kl ∂ l + d X k =1 c k ∂ k − d X k =1 ∂ k b k + c on bounded Lipschitz domains with non-local Robin boundary conditions. In this generalsituation it is necessary to pay special attention to the definition and properties of theco-normal and adjoint co-normal derivative, and to the properties of the correspondingsesquilinear forms and operators. Furthermore, the unique solvability of the homogeneousand inhomogeneous Dirichlet boundary value problems is discussed. For the convenience ofthe reader we provide proofs of these preparatory results in Section 3. Our main result onthe characterization of Jordan chains of second-order elliptic partial differential operatorswith local or non-local Robin boundary conditions via Jordan chains of the Dirichlet-to-Neumann map λ D ( λ ) and the boundary operator B is formulated and proved inSection 4. The proof is technical and requires the preparatory Lemma 4.5. Finally, inSubsection 5.1 we discuss a more regular situation in which the bounded domain Ω isassumed to have a C -smooth boundary and the coefficients of the elliptic operator areslightly more regular. In this setting one then obtains a Dirichlet-to-Neumann operatoracting from H / (Γ) into H / (Γ) and a variant of Theorem 4.1 for H (Ω)-smooth Jordan3hains. In Subsection 5.2 we reconsider the Dirichlet-to-Neumann operator on a Lipschitzdomain, but now we treat the Dirichlet-to-Neumann operator acting from H (Γ) into L (Γ).For this we require a smoothness and symmetry condition on the principal coefficients. Acknowledgements.
J. Behrndt is most grateful for the stimulating research stay andthe hospitality at the University of Auckland, where parts of this paper were written.This work is supported by the Austrian Science Fund (FWF), project P 25162-N26 andpart of this work is supported by the Marsden Fund Council from Government funding,administered by the Royal Society of New Zealand.
Throughout this paper the field is the complex numbers. Let A be a linear operator in aBanach space H . Further, let k ∈ N , f , . . . , f k ∈ H and λ ∈ C . Then we say that thevectors { f , . . . , f k } form a Jordan chain for A at λ if f j ∈ dom A for all j ∈ { , . . . , k } satisfy ( A − λ ) f j = f j − for all j ∈ { , . . . , k } with f = 0 and we set f − = 0. The vector f is called an eigen-vector of A at the eigenvalue λ and the vectors f , . . . , f k are said to be generalizedeigenvectors of A at λ . Note that the generalized eigenvectors are all nonzero.The notion of Jordan chains exists also for holomorphic operator functions and goesback to the work of M.V. Keldysh [Kel], for more details we also refer the reader to themonograph [Mark, § H and H be Banach spaces, O ⊂ C an open set and for all λ ∈ O let M ( λ ) ∈ L ( H , H ). Assume, in addition, that the operator function λ M ( λ )is holomorphic on O and denote the l -th derivative of M ( · ) at λ ∈ O by M ( l ) ( λ ). Let k ∈ N and ϕ , . . . , ϕ k ∈ H . Then we say that the vectors { ϕ , . . . , ϕ k } form a Jordanchain for the function M ( · ) at λ ∈ O if j X l =0 l ! M ( l ) ( λ ) ϕ j − l = 0for all j ∈ { , . . . , k } and ϕ = 0. The vector ϕ is called an eigenvector of the operatorfunction M ( · ) at the eigenvalue λ and the vectors ϕ , . . . , ϕ k are said to be generalizedeigenvectors of M ( · ) at λ .Observe that in the special case H = H and C ∈ L ( H ) the notion of Jordan chainfor the operator C at λ ∈ C and the notion of Jordan chain for the function λ C − λ at λ ∈ C coincide. 4 Elliptic differential operators and Dirichlet-to-Neu-mann maps
Let Ω ⊂ R d be a bounded Lipschitz domain with boundary Γ. By H (Ω) we denote the L -based Sobolev space of order 1 on Ω and H (Ω) denotes the closure of the compactlysupported C ∞ c (Ω)-functions in H (Ω). On the Lipschitz boundary Γ the Sobolev space H / (Γ) of order 1 / H − / (Γ) and h· , ·i stands for the extension of the L (Γ) inner product onto the pair H / (Γ) × H − / (Γ). Recallfrom [McL] Theorem 3.37 that there is a continuous trace map Tr : H (Ω) → H / (Γ)such that Tr f = f | Γ for all f ∈ H (Ω) ∩ C (Ω) and it admits a bounded right inverse.For all k, l ∈ { , . . . , d } fix c kl , b k , c k , c ∈ L ∞ (Ω). We recall that the field is the complexnumbers, so we emphasise that all coefficients are complex valued. Assume that there existsa µ > d X k,l =1 c kl ( x ) ξ k ξ l ≥ µ | ξ | for all x ∈ Ω and ξ ∈ C d . Define the sesquilinear form a : H (Ω) × H (Ω) → C by a ( f, g ) = d X k,l =1 Z Ω c kl ( ∂ l f ) ∂ k g + d X k =1 Z Ω c k ( ∂ k f ) g + d X k =1 Z Ω b k f ∂ k g + Z Ω c f g. The form a is continuous in the sense that there exists an M ≥ | a ( f, g ) | ≤ M k f k H (Ω) k g k H (Ω) for all f, g ∈ H (Ω). One verifies in the same way as in the proof of[AE1] Lemma 3.7 that the form is elliptic and hence [AE3] Lemma 3.1 implies that a is aclosed sectorial form.Introduce A : H (Ω) → ( H (Ω)) ∗ by h A f, g i ( H (Ω)) ∗ × H (Ω) = a ( f, g ) . In order to introduce the co-normal derivative we need a lemma. Note that the ellipticitycondition on the principal coefficients is not needed in the next lemma.
Lemma 3.1.
Let f ∈ H (Ω) and suppose that A f ∈ L (Ω) . Then there exists a unique ψ ∈ H − / (Γ) such that a ( f, g ) − ( A f, g ) L (Ω) = h ψ, Tr g i H − / (Γ) × H / (Γ) for all g ∈ H (Ω) . Moreover, there exists a constant c > , independent of f , such that k ψ k H − / (Γ) ≤ c ( k f k H (Ω) + k A f k L (Ω) ) . Proof . Define F : H (Ω) → C by F ( g ) = a ( f, g ) − ( A f, g ) L (Ω) . Then F is anti-linear andbounded. Explicitly, there exists an M ≥
0, independent of f , such that k F k H (Ω) ∗ ≤ M k f k H (Ω) + k A f k L (Ω) . F ( g ) = 0 for all g ∈ H (Ω). Hence there exists a unique anti-linear e F : H / (Γ) → C such that e F (Tr g ) = F ( g ) for all g ∈ H (Ω). The map e F is bounded and k e F k H / (Γ) ∗ ≤k F k H (Ω) ∗ k Z k , where Z : H / (Γ) → H (Ω) is a bounded right inverse of Tr . Write ψ = e F ∈ H / (Γ) ∗ = H − / (Γ). Then e F ( ϕ ) = h ψ, ϕ i H − / (Γ) × H / (Γ) for all ϕ ∈ H / (Γ)and the lemma follows.If f ∈ H (Ω) with A f ∈ L (Ω), then we denote by γ N f ∈ H − / (Γ) the function suchthat a ( f, g ) − ( A f, g ) L (Ω) = h γ N f, Tr g i H − / (Γ) × H / (Γ) for all g ∈ H (Ω). We call γ N f the co-normal derivative of f .Denote by a D the restriction of a to H (Ω) × H (Ω). Then a D is a continous elliptic formand hence a closed sectorial form (cf. [AE3] Lemma 3.1.) Denote by A D the m-sectorialoperator associated with the form a D . It follows that A D is the Dirichlet realization of A in L (Ω) given by A D f = A f, dom A D = (cid:8) f ∈ H (Ω) : A f ∈ L (Ω) (cid:9) . Lemma 3.2.
Let λ ∈ ρ ( A D ) . Then the following assertions hold. (a) For all ϕ ∈ H / (Γ) there exists a unique solution f ∈ H (Ω) of the homogeneousboundary value problem ( A − λ ) f = 0 and Tr f = ϕ. (3.1) Moreover, the map ϕ f is continuous from H / (Γ) into H (Ω) . (b) For all ϕ ∈ H / (Γ) and all h ∈ L (Ω) there exists a unique solution f ∈ H (Ω) ofthe inhomogeneous boundary value problem ( A − λ ) f = h and Tr f = ϕ. (3.2) Proof . ‘(a)’. The existence follows as in the proof of [AE6] Lemma 2.1. For completenesswe give the details. There exists a T ∈ L ( H (Ω)) such that( T f, g ) H (Ω) = a D ( f, g ) − λ ( f, g ) L (Ω) for all f, g ∈ H (Ω). Further there exists an ω > b : H (Ω) × H (Ω) → C given by b ( f, g ) = a D ( f, g ) − λ ( f, g ) L (Ω) + ω ( f, g ) L (Ω) is coercive.Let j : H (Ω) → L (Ω) be the (compact) inclusion map. Then b ( f, g ) = (( T + K ) f, g ) H (Ω) for all f, g ∈ H (Ω), where K = ωj ∗ j . So T + K is invertible by the Lax–Milgram theorem.Consequently T is a Fredholm operator because K is compact. Now T is injective since λ ∈ ρ ( A D ). Hence T is surjective.There exists an f ∈ H (Ω) such that Tr f = ϕ . Hence there exists an h ∈ H (Ω) suchthat ( T h, g ) H (Ω) = a ( f , g ) − λ ( f , g ) L (Ω) for all g ∈ H (Ω). Then f = f − h satisfies h A f − λf, g i ( H (Ω)) ∗ × H (Ω) = a ( f , g ) − λ ( f , g ) L (Ω) − a D ( h, g ) + λ ( h, g ) L (Ω) = 06nd hence ( A − λ ) f = 0. The uniqueness is easy. The continuity of the map follows fromthe closed graph theorem.‘(b)’. By Statement (a) there exists an f ∈ H (Ω) such that ( A − λ ) f = 0 andTr f = ϕ . Then f + ( A D − λ ) − h is a solution to the problem (3.2). Again the uniquenessis easy.Let λ ∈ ρ ( A D ). Now we are able to define the Dirichlet-to-Neumann operator D ( λ ) : H / (Γ) → H − / (Γ). Let ϕ ∈ H / (Γ). By Lemma 3.2(a) there exists a uniquesolution f ∈ H (Ω) of the homogeneous boundary value problem (3.1). Then A f = λf ∈ L (Ω). Hence one can define D ( λ ) ϕ = γ N f. Then D ( λ ) is bounded operator from H / (Γ) into H − / (Γ) by the last parts of Lemmas 3.1and 3.2(a).We need two holomorphy results. Lemma 3.3. (a)
Let ϕ ∈ H / (Γ) . For all λ ∈ ρ ( A D ) let g λ ∈ H (Ω) be the unique element such that ( A − λ ) g λ = 0 and Tr g λ = ϕ . Then the map λ g λ is holomorphic from ρ ( A D ) into H (Ω) . (b) The map λ D ( λ ) is holomorphic from ρ ( A D ) into L ( H / (Γ) , H − / (Γ)) . Proof . ‘(a)’. Fix λ ∈ ρ ( A D ). By Lemma 3.2(a) there exists a unique g λ ∈ H (Ω) suchthat ( A − λ ) g λ = 0 and Tr g λ = ϕ . Let λ ∈ ρ ( A D ) and consider g = (cid:0) λ − λ )( A D − λ ) − (cid:1) g λ ∈ H (Ω) . (3.3)Then ( A − λ ) g = ( A − λ ) g λ + ( λ − λ ) g λ = 0 and Tr g = Tr g λ = ϕ . Since the solutionof the homogeneous boundary value problem ( A − λ ) f = 0 with Tr f = ϕ , is unique byLemma 3.2(a) it follows that g = g λ . Now the holomorphy of the resolvent λ ( A D − λ ) − in (3.3) implies that the map λ g λ is holomorphic from ρ ( A D ) into H (Ω).‘(b)’. Let ϕ ∈ H / (Γ) and h ∈ H (Ω). For all λ ∈ ρ ( A D ) let g λ ∈ H (Ω) be as inStatement (a). Then h D ( λ ) ϕ, Tr h i H − / (Γ) × H / (Γ) = h γ N g λ , Tr h i H − / (Γ) × H / (Γ) = a ( g λ , h ) − ( A g λ , h ) L (Ω) = a ( g λ , h ) − λ ( g λ , h ) L (Ω) for all λ ∈ ρ ( A D ). Since λ g λ is holomorphic from ρ ( A D ) into H (Ω) by Statement (a),it follows that λ D ( λ ) is holomorphic with respect to the weak operator topology on L ( H / (Γ) , H − / (Γ)), and therefore it is also holomorphic with respect to the uniformoperator topology. 7or all l ∈ N we denote the l -th derivative of λ D ( λ ) at λ ∈ ρ ( A D ) by D ( l ) ( λ ). Thenaccording to Lemma 3.3(b) one has D ( l ) ( λ ) ∈ L ( H / (Γ) , H − / (Γ))for all λ ∈ ρ ( A D ).The dual form a ∗ of a is defined by dom ( a ∗ ) = H (Ω) and a ∗ ( f, g ) = a ( g, f ) for all f, g ∈ H (Ω). So a ∗ ( f, g ) = d X k,l =1 Z Ω c lk ( ∂ l f ) ∂ k g + d X k =1 Z Ω b k ( ∂ k f ) g + d X k =1 Z Ω c k f ∂ k g + Z Ω c f g. Obviously a ∗ is of the same type as a , with c kl replaced by c lk , etc. Similar to the definitionof A with respect to a , we can define the operator e A : H (Ω) → ( H (Ω)) ∗ by h e A f, g i ( H (Ω)) ∗ × H (Ω) = a ∗ ( f, g ) . As in Lemma 3.1 it follows that for all f ∈ H (Ω) with e A f ∈ L (Ω), there exists a unique e γ N f ∈ H − / (Γ) such that a ∗ ( f, g ) − ( e A f, g ) L (Ω) = h e γ N f, Tr g i H − / (Γ) × H / (Γ) for all g ∈ H (Ω). Using all definitions it is easy to prove the following version of Green’ssecond identity. Lemma 3.4.
Let f, g ∈ H (Ω) and suppose that A f, e A g ∈ L (Ω) . Then ( A f, g ) L (Ω) − ( f, e A g ) L (Ω) = h Tr f, e γ N g i H / (Γ) × H − / (Γ) − h γ N f, Tr g i H − / (Γ) × H / (Γ) . (3.4)Denote by a ∗ D the restriction of the dual form a ∗ to H (Ω) × H (Ω). Then a ∗ D is a closedsectorial form and the m-sectorial operator associated with a ∗ D is equal to the adjoint A ∗ D of A D , see [Kat] Theorem VI.2.5. It follows that A ∗ D is the Dirichlet realization of e A in L (Ω) given by A ∗ D f = e A f, dom A ∗ D = (cid:8) f ∈ H (Ω) : e A f ∈ L (Ω) (cid:9) . Similarly to the Dirichlet-to-Neumann map D ( λ ) ∈ L ( H / (Γ) , H − / (Γ)) one asso-ciates the Dirichlet-to-Neumann map e D ( λ ) ∈ L ( H / (Γ) , H − / (Γ)) to the adjoint form a ∗ for all λ ∈ ρ ( A ∗ D ). A simple computation based on Greens second identity (3.4) shows h D ( λ ) ϕ, ψ i H − / (Γ) × H / (Γ) = h ϕ, e D ( λ ) ψ i H / (Γ) × H − / (Γ) (3.5)for all ϕ, ψ ∈ H / (Γ) and λ ∈ ρ ( A D ).Finally we introduce the Robin operator. Let B ∈ L ( H / (Γ) , H − / (Γ)). We assumethat there is an η > h Bϕ, ϕ i H − / (Γ) × H / (Γ) ≤ η k ϕ k L (Γ) (3.6)8or all ϕ ∈ H / (Γ). Note that the restriction to the space H / (Γ) of every boundedoperator B in L (Γ) can be viewed as an operator in L ( H / (Γ) , H − / (Γ)) that satisfies(3.6). We also note that the above assumption on B ∈ L ( H / (Γ) , H − / (Γ)) can begeneralized further as in for example [GM2] Hypothesis 4.1. Next we define the sesquilinearform a B : H (Ω) × H (Ω) → C by a B ( f, g ) = a ( f, g ) − h B Tr f, Tr g i H − / (Γ) × H / (Γ) . Proposition 3.5.
The form a B is densely defined, closed and sectorial in L (Ω) . Theassociated m-sectorial operator A B f = A f, dom A B = (cid:8) f ∈ H (Ω) : A f ∈ L (Ω) and γ N f = B Tr f (cid:9) , is the Robin realisation of A in L (Ω) . Proof . We will show first that a B is elliptic, that is, there are ν ∈ R and µ > a B ( f ) + ν k f k L (Ω) ≥ µ k f k H (Ω) (3.7)for all f ∈ H (Ω). Clearly there are µ , ω > a ( f ) ≥ µ k f k H (Ω) − ω k f k L (Ω) for all f ∈ H (Ω) (cf. [AE1] Lemma 3.7.) Choose ε < µ η , where η > H (Ω) → L (Γ) there exists a c > k Tr f k L (Γ) ≤ ε k f k H (Ω) + c k f k L (Ω) for all f ∈ H (Ω). ThenRe h B Tr f, Tr f i H − / (Γ) × H / (Γ) ≤ η k Tr f k L (Γ) ≤ µ k f k H (Ω) + ηc k f k L (Ω) and henceRe a B ( f ) = Re a ( f ) − Re h B Tr f, Tr f i H − / (Γ) × H / (Γ) ≥ µ k f k H (Ω) − ( ω + ηc ) k f k L (Ω) for all f ∈ H (Ω). So (3.7) holds with µ = µ and ν = ω + ηc , therefore a B is elliptic.Hence a B is a densely defined, closed, sectorial form (see [AE3] Lemma 3.1).The graph of the m-sectorial operator associated to a B is given by G = (cid:8) ( f, h ) ∈ H (Ω) × L (Ω) : a B ( f, g ) = ( h, g ) L (Ω) for all g ∈ H (Ω) (cid:9) and it remains to show that G coincides with the Robin realisation A B . Now let f ∈ dom G and write h = Gf ∈ L (Ω). Then f ∈ H (Ω) and h A f, g i ( H (Ω)) ∗ × H (Ω) = a ( f, g ) = a B ( f, g ) = ( h, g ) L (Ω) for all g ∈ H (Ω). So A f = h = Gf ∈ L (Ω). If g ∈ H (Ω), then a ( f, g ) − ( A f, g ) L (Ω) = a B ( f, g ) + h B Tr f, Tr g i H − / (Γ) × H / (Γ) − ( h, g ) L (Ω) = h B Tr f, Tr g i H − / (Γ) × H / (Γ) . So γ N f = B Tr f and hence f ∈ dom A B . The converse inclusion follows similarly.9 Jordan chains of Robin realizations
Adopt the assumptions and notation as in Section 3. In this section we formulate andprove our main result on the characterization of Jordan chains of the m-sectorial Robinrealization A B of A via the operator function λ D ( λ ) − B . Our goal is to show thefollowing theorem. Theorem 4.1.
Let A B be the Robin realisation of A in L (Ω) as in Proposition 3.5, let λ ∈ ρ ( A D ) and consider the holomorphic function λ D ( λ ) − B (4.1) from ρ ( A D ) into L ( H / (Γ) , H − / (Γ)) . Then the following holds. (a) Let { f , . . . , f k } be a Jordan chain for A B at λ . For all m ∈ { , . . . , k } define ϕ m = Tr f m . Then { ϕ , . . . , ϕ k } is a Jordan chain for the function (4.1) at λ . (b) Let { ϕ , . . . , ϕ k } be a Jordan chain for the function (4.1) at λ . Set f − = 0 . For all m ∈ { , . . . , k } let f m ∈ H (Ω) be the unique solution of the boundary value problem ( A − λ ) f m = f m − , Tr f m = ϕ m . Then { f , . . . , f k } is a Jordan chain for A B at λ . For the special case k = 0 one obtains the following well-known result. Corollary 4.2.
Adopt the notation and assumptions as in Theorem 4.1. Then the followingholds. (a) If f is an eigenvector of A B at λ , then D ( λ )Tr f = B Tr f and Tr f = 0 . (b) If D ( λ ) ϕ = Bϕ and ϕ = 0 , then the unique solution f ∈ H (Ω) of the boundaryvalue problem ( A − λ ) f = 0 , Tr f = ϕ , is an eigenvector of A B at λ . Corollary 4.3.
Adopt the notation and assumptions as in Theorem 4.1. Then
Tr (ker ( A B − λ )) = ker ( D ( λ ) − B ) and Tr is a bijection from ker ( A B − λ ) onto ker ( D ( λ ) − B ) . Remark 4.4.
We can mention here that the assumption λ ∈ ρ ( A D ) in Theorem 4.1 andCorollary 4.2 is really needed. In fact, one may define the Dirichlet-to-Neumann graph as alinear relation consisting of the Cauchy data for all λ ∈ σ p ( A D ). By [Fil] Theorem 1 thereexist µ > λ ∈ R , u ∈ C ∞ c ( R ) \ { } and a H¨older continuous function g : R → [ µ, ∞ )such that − div g ∇ u = λu . Let Ω be a Lipschitz domain with supp u ⊂ Ω. Choose c kl = g | Ω δ kl , b k = c k = c = 0 for all k, l ∈ { , . . . , d } and f = u | Ω . Let B ∈ L ( L (Γ)).Then f is an eigenfunction of A B at λ . But Tr f = 0. So one cannot drop the assumption λ ∈ ρ ( A D ) in Corollary 4.2(a). 10bserve that the homogenenous and inhomogeneous boundary value problems in The-orem 4.1(b) and Corollary 4.2(b) admit unique solutions by Lemma 3.2. The proof ofTheorem 4.1 requires quite some preparation. The next lemma is particularly useful; itsproof is partly based on an argument that was given by V.A. Derkach for symmetric andselfadjoint linear relations in Krein spaces; see also [DM3] Section 7.4.4. Lemma 4.5.
Let A B be the Robin realisation of A in L (Ω) as in Proposition 3.5 andlet { f , . . . , f k } be a Jordan chain of A B at λ ∈ ρ ( A D ) . For all m ∈ { , . . . , k } define ϕ m = Tr f m ∈ H / (Γ) . Let ϕ ∈ H / (Γ) and let g ∈ H (Ω) be the unique solution of theadjoint problem ( e A − λ ) g = 0 such that Tr g = ϕ . Then the following holds. (a) If j ∈ { , . . . , k } , then ( f j − , g ) L (Ω) = h D ( λ ) ϕ j − Bϕ j , ϕ i H − / (Γ) × H / (Γ) . (4.2)(b) If j ∈ { , . . . , k + 1 } , then ( f j − , g ) L (Ω) = − j X l =1 l ! h D ( l ) ( λ ) ϕ j − l , ϕ i H − / (Γ) × H / (Γ) . (4.3) Proof . For all λ ∈ ρ ( A D ) let g λ ∈ H (Ω) be the unique solution of the adjoint problem( e A − λ ) g λ = 0 such that Tr g λ = ϕ ; see Lemma 3.2(a). Then g λ = g . We set f − = 0.‘(a)’. If j ∈ { , . . . , k } and λ ∈ ρ ( A D ), then f j ∈ dom A B , so A B f j = A f j and γ N f j = B Tr f j by Proposition 3.5. Therefore( A B f j , g λ ) L (Ω) − ( f j , λg λ ) L (Ω) = ( A f j , g λ ) L (Ω) − ( f j , e A g λ ) L (Ω) = h Tr f j , e γ N g λ i H / (Γ) × H − / (Γ) − h γ N f j , Tr g λ i H − / (Γ) × H / (Γ) = h Tr f j , e D ( λ )Tr g λ i H / (Γ) × H − / (Γ) − h B Tr f j , Tr g λ i H − / (Γ) × H / (Γ) = (cid:10) D ( λ ) ϕ j − Bϕ j , ϕ (cid:11) H − / (Γ) × H / (Γ) , (4.4)where we used (3.5) in the last step. Choosing λ = λ gives( f j − , g ) L (Ω) = (cid:0) ( A B − λ ) f j , g λ (cid:1) L (Ω) = ( A B f j , g λ ) L (Ω) − ( f j , λ g λ ) L (Ω) = (cid:10) D ( λ ) ϕ j − Bϕ j , ϕ (cid:11) H − / (Γ) × H / (Γ) , which proves (4.2). Note that j = 0 gives h D ( λ ) ϕ − Bϕ , ϕ i H − / (Γ) × H / (Γ) = 0 and hence D ( λ ) ϕ = Bϕ . (4.5)‘(b)’. We shall show that − ( f j − , g λ ) L (Ω) = j X l =1 * λ − λ ) l D ( λ ) − l − X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ j − l , ϕ + H − / (Γ) × H / (Γ) (4.6)11or all j ∈ { , . . . , k + 1 } and λ ∈ ρ ( A D ) \ { λ } . Once we have shown this, then the equality(4.3) easily follows by taking the limit λ → λ . In fact, the left hand side of (4.6) tends to − ( f j − , g λ ) L (Ω) = − ( f j − , g ) L (Ω) by Lemma 3.3(a), and using the Taylor expansion D ( λ ) = ∞ X s =0 s ! ( λ − λ ) s D ( s ) ( λ )it is easy to see that for λ → λ the right hand side in (4.6) tends to j X l =1 l ! h D ( l ) ( λ ) ϕ j − l , ϕ i H − / (Γ) × H / (Γ) . We prove formula (4.6) by induction. If j = 1 and λ ∈ ρ ( A D ) \ { λ } , then (4.4) gives − ( λ − λ ) L (Ω) ( f , g λ ) L (Ω) = ( λ f , g λ ) L (Ω) − ( f , λg λ ) L (Ω) = ( A B f , g λ ) L (Ω) − ( f , λg λ ) L (Ω) = (cid:10) D ( λ ) ϕ − Bϕ , ϕ (cid:11) H − / (Γ) × H / (Γ) = (cid:10) ( D ( λ ) − D ( λ )) ϕ , ϕ (cid:11) H − / (Γ) × H / (Γ) , where we used (4.5) in the last step. So (4.6) is valid if j = 1.Let m ∈ { , . . . , k } and suppose that (4.6) is valid for j = m . Then by taking the limit λ → λ one deduces that − ( f m − , g ) L (Ω) = m X l =1 l ! h D ( l ) ( λ ) ϕ m − l , ϕ i H − / (Γ) × H / (Γ) , and together with (4.2) we conclude (cid:10) D ( λ ) ϕ m − Bϕ m , ϕ (cid:11) H − / (Γ) × H / (Γ) = − m X l =1 l ! h D ( l ) ( λ ) ϕ m − l , ϕ i H − / (Γ) × H / (Γ) . (4.7)Now let us prove the formula (4.6) for j = m + 1. Let λ ∈ ρ ( A D ) \ { λ } . Then a simplecomputation shows m +1 X l =1 λ − λ ) l D ( λ ) − l − X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ m +1 − l = m +1 X l =2 λ − λ ) l D ( λ ) − l − X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ m +1 − l + D ( λ ) − D ( λ ) λ − λ ϕ m = m X l =1 λ − λ ) l +1 D ( λ ) − l X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ m − l + D ( λ ) − D ( λ ) λ − λ ϕ m
12 1 λ − λ m X l =1 λ − λ ) l D ( λ ) − l − X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ m − l − λ − λ m X l =1 l ! D ( l ) ( λ ) ϕ m − l + D ( λ ) − D ( λ ) λ − λ ϕ m and using (4.6) for j = m for the first term on the right hand side, and (4.7) for the secondterm on the right hand side gives m +1 X l =1 * λ − λ ) l D ( λ ) − l − X s =0 s ! ( λ − λ ) s D ( s ) ( λ ) ! ϕ m +1 − l , ϕ + H − / (Γ) × H / (Γ) = − λ − λ ( f m − , g λ ) L (Ω) + 1 λ − λ (cid:10) D ( λ ) ϕ m − Bϕ m , ϕ (cid:11) H − / (Γ) × H / (Γ) + 1 λ − λ (cid:10) D ( λ ) ϕ m − D ( λ ) ϕ m , ϕ (cid:11) H − / (Γ) × H / (Γ) = 1 λ − λ (cid:10) D ( λ ) ϕ m − Bϕ m , ϕ (cid:11) H − / (Γ) × H / (Γ) − λ − λ ( f m − , g λ ) L (Ω) = 1 λ − λ (cid:0) ( A B f m , g λ ) L (Ω) − ( f m , λg λ ) L (Ω) − ( f m − , g λ ) (cid:1) L (Ω) = 1 λ − λ (cid:0) ( f m − + λ f m , g λ ) L (Ω) − ( f m , λg λ ) L (Ω) − ( f m − , g λ ) (cid:1) L (Ω) = − ( f m , g λ ) L (Ω) , where (4.4) was used for j = m in third equality and ( A B − λ ) f m = f m − was used in thefourth equality. We have shown (4.6) for j = m + 1. The proof of (b) is complete.Now we are able to prove the main theorem. Proof of Theorem 4.1. ‘(a)’. Let { f , . . . , f k } form a Jordan chain for A B at λ ∈ ρ ( A D )and let ϕ j = Tr f j ∈ H / (Γ) for all j ∈ { , . . . , k } be the corresponding traces. We haveto prove that j X l =0 l ! D ( l ) ( λ ) ϕ j − l = Bϕ j (4.8)for all j ∈ { , . . . , k } and that ϕ = 0.Using Proposition 3.5 it is easy to see that D ( λ ) ϕ − Bϕ = D ( λ )Tr f − B Tr f = γ N f − γ N f = 0and hence (4.8) is valid if j = 0. Furthermore, ϕ = Tr f = 0 as otherwise f ∈ dom A D and therefore ( A D − λ ) f = ( A B − λ ) f = 0, which together with λ ∈ ρ ( A D ) wouldimply f = 0.Let j ∈ { , . . . , k } and let ϕ ∈ H / (Γ). Then Lemma 4.5 gives (cid:10) D ( λ ) ϕ j − Bϕ j , ϕ (cid:11) H − / (Γ) × H / (Γ) = − j X l =1 l ! h D ( l ) ( λ ) ϕ j − l , ϕ i H − / (Γ) × H / (Γ) . Bϕ j = D ( λ ) ϕ j + j X l =1 l ! D ( l ) ( λ ) ϕ j − l = j X l =0 l ! D ( l ) ( λ ) ϕ j − l as required.‘(b)’. Assume that { ϕ , . . . , ϕ k } form a Jordan chain of the function λ D ( λ ) − B at λ , that is, (4.8) is valid for all j ∈ { , , . . . k } and ϕ = 0. In the following we construct aJordan chain { f , . . . , f k } of A B at λ such that the corresponding traces are given by theset of vectors { ϕ , . . . , ϕ k } . We proceed by induction. According to Lemma 3.2(a) thereexists a unique f ∈ H (Ω) such that ( A − λ ) f = 0 and Tr f = ϕ . Making use of (4.8)for j = 0 we obtain γ N f = D ( λ )Tr f = D ( λ ) ϕ = Bϕ = B Tr f and hence f ∈ dom A B with ( A B − λ ) f = 0 by Proposition 3.5. Since ϕ = 0 it is clearthat also f = 0.Now let m ∈ { , . . . , k } and assume that there are f , . . . , f m − ∈ H (Ω) such that ϕ j = Tr f j for all j ∈ { , . . . , m − } and the vectors { f , . . . , f m − } form a Jordan chainfor A B at λ . By Lemma 3.2(b) there exists a unique vector f m ∈ H (Ω) such that( A − λ ) f m = f m − and Tr f m = ϕ m . (4.9)We shall prove that γ N f m = B Tr f m . Once we proved that, it follows that f m ∈ dom A B and ( A B − λ ) f m = f m − .By assumption and (4.9) one deduces that D ( λ )Tr f m = D ( λ ) ϕ m = Bϕ m − m X l =1 l ! D ( l ) ( λ ) ϕ m − l = B Tr f m − m X l =1 l ! D ( l ) ( λ ) ϕ m − l . Let ϕ ∈ H / (Γ). By Lemma 3.2(a) there exists a unique g ∈ H (Ω) such that ( e A − λ ) g = 0and Tr g = ϕ . Then(( A − λ ) f m , g ) L (Ω) = ( A f m , g ) L (Ω) − ( f m , λ g ) L (Ω) = ( A f m , g ) L (Ω) − ( f m , e A g ) L (Ω) = h Tr f m , e γ N g i H / (Γ) × H − / (Γ) − h γ N f m , Tr g i H − / (Γ) × H / (Γ) = h Tr f m , e D ( λ )Tr g i H / (Γ) × H − / (Γ) − h γ N f m , Tr g i H − / (Γ) × H / (Γ) = (cid:10) D ( λ )Tr f m − γ N f m , ϕ (cid:11) H − / (Γ) × H / (Γ) = * B Tr f m − γ N f m − m X l =1 l ! D ( l ) ( λ ) ϕ m − l , ϕ + H − / (Γ) × H / (Γ) . On the other hand, as { f , . . . , f m − } is a Jordan chain of A B at λ we have(( A − λ ) f m , g ) L (Ω) = ( f m − , g ) L (Ω) = − m X l =1 l ! h D ( l ) ( λ ) ϕ m − l , ϕ i H − / (Γ) × H / (Γ)
14y Lemma 4.5(b). Therefore h B Tr f m − γ N f m , ϕ i H − / (Γ) × H / (Γ) = 0 for all ϕ ∈ H / (Γ).Thus γ N f m = B Tr f m as required. So { f , . . . , f m } is a Jordan chain for A B at λ withtraces { ϕ , . . . , ϕ m } . Remark 4.6.
In the abstract setting of boundary triplets and their Weyl functions foradjoint pairs [LS, MM, Vai] it is known under a natural unique continuation hypothesis thatthe poles of the Weyl function correspond to the isolated eigenvalues of the fixed extension,see [BMNW1, Theorem 4.4]. See also [BMNW2, BHMNW, BL] for related results in thecontext of indefinite inner product spaces.
In the previous section we considered the Dirichlet-to-Neumann operator D ( λ ) : H / (Γ) → H − / (Γ) and the Jordan chain with respect to the holomorphic operator function λ D ( λ ) − B from ρ ( A D ) into L ( H / (Γ) , H − / (Γ)), where B ∈ L ( H / (Γ) , H − / (Γ)) satisfies(3.6).Except from the obvious ellipticity condition and to have a Lipschitz domain, therewere no conditions on the coefficients: merely bounded measurable and complex valued.There are two other Dirichlet-to-Neumann operators that we consider in this section. C -domains Throughout this subsection we suppose that Ω is a C -domain, c kl ∈ C (Ω) and b k = 0 forall k, l ∈ { , . . . , d } . We summarise some regularity results that we need in this subsection. Lemma 5.1. (a) If f ∈ H (Ω) , then Tr f ∈ H / (Γ) , A f ∈ L (Ω) and γ N f = d X k,l =1 ν k Tr ( c kl ∂ l f ) ∈ H / (Γ) . Moreover, the map f γ N f is continuous from H (Ω) into H / (Γ) . (b) Let λ ∈ ρ ( A D ) . For all ϕ ∈ H / (Γ) there exists a unique f ∈ H (Ω) such that ( A − λ ) f = 0 and Tr f = ϕ . Moreover, the map ϕ f is continuous from H / (Γ) into H (Ω) . (c) Let λ ∈ ρ ( A D ) . For all h ∈ L (Ω) and ϕ ∈ H / (Γ) there exists a unique f ∈ H (Ω) such that ( A − λ ) f = h and Tr f = ϕ . Proof . ‘(a)’. This follows from [Gri] Theorem 1.5.1.2 and the divergence theorem.‘(b)’. By [Gri] Theorem 1.5.1.2 there exists an f ∈ H (Ω) such that Tr f = ϕ .Then it follows [Eva] Theorem 6.3.4 that there exists a unique h ∈ H (Ω) such that( A − λ ) h = ( A − λ ) f and Tr h = 0. Therefore f = f − h satisfies the requirements.15he uniqueness is easy. The continuity follows from Lemma 3.2(a) and the closed graphtheorem.‘(c)’. This can be proved similarly.For all λ ∈ ρ ( A D ) define the Dirichlet-to-Neumann operator b D ( λ ) : H / (Γ) → H / (Γ)as follows. Let ϕ ∈ H / (Γ). By Lemma 5.1(b) there exists a unique f ∈ H (Ω) such that( A − λ ) f = 0 and Tr f = ϕ . Define b D ( λ ) ϕ = γ N f ∈ H / (Γ) by Lemma 5.1(a). Then b D ( λ ) is a bounded operator.Next we consider holomorphy. Lemma 5.2.
The map λ b D ( λ ) from ρ ( A D ) into L ( H / (Γ) , H / (Γ)) is holomorphic. Proof . For all ϕ ∈ H / (Γ) and ψ ∈ H / (Γ) define α ϕ,ψ : L ( H / (Γ) , H / (Γ)) → C by α ϕ,ψ ( F ) = ( F ϕ, ψ ) L (Γ) . Then α ϕ,ψ ∈ L ( H / (Γ) , H / (Γ)) ∗ . Let W = span { α ϕ,ψ : ϕ ∈ H / (Γ) and ψ ∈ H / (Γ) } .Since H / (Γ) is dense in L (Γ), it follows that the space W is separating, that is, if F ∈ L ( H / (Γ) , H / (Γ)) with α ( F ) = 0 for all α ∈ W , then it follows that F = 0. If ϕ ∈ H / (Γ) and ψ ∈ H / (Γ), then α ϕ,ψ ( b D ( λ )) = ( b D ( λ ) ϕ, ψ ) L (Γ) = h D ( λ ) ϕ, ψ i H − / (Γ) × H / (Γ) for all λ ∈ ρ ( A D ). Hence the map λ α ϕ,ψ ( b D ( λ )) is holomorphic for all ϕ ∈ H / (Γ)and ψ ∈ H / (Γ) by Lemma 3.3(b). Consequently the map λ b D ( λ ) from ρ ( A D ) into L ( H / (Γ) , H / (Γ)) is holomorphic by [ABHN] Theorem A.7.The alluded variation of Theorem 4.1 is as follows. Theorem 5.3.
Let B ∈ L ( H / (Γ) , H / (Γ)) and suppose there exists an η > such that Re(
Bϕ, ϕ ) L (Γ) ≤ η k ϕ k L (Γ) for all ϕ ∈ H / (Γ) . Let A B be the Robin realisation of A in L (Ω) as in Proposition 3.5,let λ ∈ ρ ( A D ) and consider the holomorphic function λ b D ( λ ) − B (5.1) from ρ ( A D ) into L ( H / (Γ) , H / (Γ)) . Then the following holds. (a) Let f , . . . , f k ∈ H (Ω) . Suppose that { f , . . . , f k } is a Jordan chain for A B at λ .For all m ∈ { , . . . , k } define ϕ m = Tr f m . Then { ϕ , . . . , ϕ k } is a Jordan chain forthe function (5.1) at λ . (b) Let { ϕ , . . . , ϕ k } be a Jordan chain for the function (5.1) at λ . Set f − = 0 . For all m ∈ { , . . . , k } let f m ∈ H (Ω) be the unique solution of the boundary value problem ( A − λ ) f m = f m − and Tr f m = ϕ m . Then { f , . . . , f k } is a Jordan chain for A B at λ . The proof is similar to the proof of Theorem 4.1, with obvious changes.16 .2 m-Sectorial operators
Throughout this subsection we merely assume again that Ω is a Lipschitz domain, butwe put conditions on the coefficients of the elliptic operator. We assume that c kl = c lk ∈ W , ∞ (Ω , R ) is real valued and b k = c k = 0 for all k ∈ { , . . . , d } . We emphasise that c can be complex valued and merely measurable. An example is the Schr¨odinger operatorwith complex potential. The Dirichlet-to-Neumann operator D ( λ ) : H / (Γ) → H − / (Γ)has been studied intensively in [BE1, BGHN, GM1, GM3]. Let D ( λ ) be the part of D ( λ )in L (Γ). So D ( λ ) ⊂ D ( λ ) and if ϕ ∈ L (Γ), then ϕ ∈ dom D ( λ ) if and only if ϕ ∈ H / (Γ)and D ( λ ) ϕ ∈ L (Γ). The operator D ( λ ) can be represented by a form. Lemma 5.4.
Let λ ∈ ρ ( A D ) . Let ϕ, ψ ∈ L (Γ) . Then the following are equivalent. (i) ϕ ∈ dom D ( λ ) and D ( λ ) ϕ = ψ . (ii) There exists an f ∈ H (Ω) such that Tr f = ϕ and a ( f, g ) − λ ( f, g ) L (Ω) = ( ψ, Tr g ) L (Γ) for all g ∈ H (Ω) . The easy proof is left to the reader.It seems that the domain of D ( λ ) depends on λ . This is not the case because of therestriction on the principal part of the elliptic operator. We collect the main properties ofthe operator D ( λ ) in the next proposition. Proposition 5.5. (a) If λ ∈ ρ ( A D ) , then the operator D ( λ ) is m-sectorial. (b) If λ ∈ ρ ( A D ) , then dom D ( λ ) = H (Ω) . (c) The map λ D ( λ ) from ρ ( A D ) into L ( H (Γ) , L (Γ)) is holomorphic. Proof . ‘(a)’. See [Ouh] Corollary 2.3.‘(b)’. ‘ ⊂ ’. Let ϕ ∈ dom D ( λ ). Then there exists an f ∈ H (Ω) such that ϕ = Tr f and ( A − λ ) f = 0. So A f = λf ∈ L (Ω) and γ N f = D ( λ ) ϕ ∈ L (Γ). Therefore [McL]Theorem 4.24(ii) implies that ϕ = Tr f ∈ H (Γ).‘ ⊃ ’. Let ϕ ∈ H (Γ). By Lemma 3.2(a) there exists a unique f ∈ H (Ω) such that( A − λ ) f = 0 and Tr f = ϕ . Then A f = λf ∈ L (Ω). Hence [McL] Theorem 4.24(i) gives γ N f ∈ L (Γ). So ϕ ∈ dom D ( λ ).‘(c)’. For all ϕ ∈ H (Γ) and ψ ∈ H / (Γ) define α ϕ,ψ : L ( H (Γ) , L (Γ)) → C by α ϕ,ψ ( F ) = ( F ϕ, ψ ) L (Γ) . Then argue as in the proof of Lemma 5.2.Now we are able to formulate another version of Theorem 4.1.17 heorem 5.6.
Let B ∈ L ( H (Γ) , L (Γ)) and suppose that there exists an η > such that Re(
Bϕ, ϕ ) L (Γ) ≤ η k ϕ k L (Γ) for all ϕ ∈ H (Γ) . Let A B be the Robin realisation of A in L (Ω) as in Proposition 3.5,let λ ∈ ρ ( A D ) and consider the holomorphic function λ b D ( λ ) − B from ρ ( A D ) into L ( H (Ω) , L (Γ)) . Then the following holds. (a) Let { f , . . . , f k } be a Jordan chain for A B at λ . For all m ∈ { , . . . , k } define ϕ m = Tr f m . Then { ϕ , . . . , ϕ k } is a Jordan chain for the function (5.1) at λ . (b) Let { ϕ , . . . , ϕ k } be a Jordan chain for the function (5.1) at λ . Set f − = 0 . For all m ∈ { , . . . , k } let f m ∈ H (Ω) be the unique solution of the boundary value problem ( A − λ ) f m = f m − and Tr f m = ϕ m . Then { f , . . . , f k } is a Jordan chain for A B at λ . The proof is similar to the proof of Theorem 4.1, with obvious changes.
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