Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps

Let $\Omega \subset {\bf R}^d$ be a bounded open set with Lipschitz boundary $\Gamma$. It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in $L_2(\Omega)$ can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from $H^{1/2}(\Gamma)$ into $H^{-1/2}(\Gamma)$. This result extends the Birman--Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.


Introduction
The Dirichlet-to-Neumann map is an important object in the analysis of elliptic partial differential equations since it can be used to describe the spectra of the associated elliptic operators. The principal strategy and advantage is that a spectral problem for a partial differential operator on a domain Ω is reduced to a spectral problem for an operator function on the boundary Γ of this domain, where, very roughly speaking, the Dirichlet and Neumann data can be measured. This type of approach to problems in spectral and scattering theory for elliptic partial differential operators was used in the self-adjoint case in, 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 the reader 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 eigenvalues of elliptic operators. To motivate our investigations let us consider here in the introduction only the special case of a Schrödinger operator A = −∆ + V on a bounded Lipschitz domain Ω ⊂ R d with d ≥ 2 and with a complex-valued potential V ∈ L ∞ (Ω). Later in this paper much more general second-order partial differential expressions A with measurable 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) : where f λ ∈ H 1 (Ω) is such that Af λ = λf λ . Here Tr f λ ∈ H 1/2 (Γ) and γ N f λ ∈ H −1/2 (Γ) denote the Dirichlet and Neumann trace of f λ , respectively, and λ ∈ C is not an eigenvalue of the Dirichlet realization A D of −∆ + V . Assume for simplicity that B : L 2 (Γ) → L 2 (Γ) is a bounded operator and consider the (non-local) Robin realization of −∆ + V defined by Ω) : γ N f = BTr f and − ∆f + V f ∈ L 2 (Ω) .
(1.1) Note that the resolvents of A D and A B are both compact operators in L 2 (Ω) due to the compactness of the embedding H 1 (Ω) → L 2 (Ω) and hence the spectra of A D and A B are discrete. It is well-known and easy to see that for all λ 0 ∈ σ p (A D ) one has λ 0 ∈ σ p (A B ) if and only if ker (D(λ 0 ) − B) = {0}. Sometimes this is referred to as a variant of the Birman-Schwinger principle. In fact, if λ 0 ∈ σ p (A B ) and f 0 ∈ dom A B is a corresponding eigenfunction, then Tr f 0 = 0 (as otherwise f 0 would be an eigenfunction for A D at λ 0 ) and In the situation where the potential V is not real-valued or the Robin boundary operator B is not symmetric the Schrödinger operator A B in (1.1) is m-sectorial, but not self-adjoint in L 2 (Ω). Therefore, in general, the eigenvalues of A B are not semisimple and besides an eigenvector f 0 also (finitely many) generalized eigenvectors f 1 , . . . , f k are associated to an eigenvalue λ 0 , which form a so-called Jordan chain. It is the main objective of the present paper to analyse the Jordan chains f 0 , f 1 , . . . , f k corresponding to an eigenvalue λ 0 of A B with the help of the Dirichlet-to-Neumann operator in a similar form as in the above mentioned Birman-Schwinger principle. In fact, using the notion of Jordan chains for holomorphic operator functions due to M.V. Keldysh [Kel] (see also [Mark,§11]), it turns out in our main result Theorem 4.1 that {f 0 , f 1 , . . . , f k } form a Jordan chain of A B at λ 0 ∈ σ p (A B ) ∩ ρ(A D ) if and only if the corresponding traces ϕ 0 = Tr f 0 , ϕ 1 = Tr f 1 , . . . , ϕ k = Tr f k form a Jordan chain for the holomorphic for all j ∈ {0, . . . , k}, where M (l) (λ 0 ) denotes the l-th derivative of the function M at λ 0 . Note that for j = 0 the characterization of the eigenvector f 0 in the Birman-Schwinger principle 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 of Jordan chains for operators and holomorphic operator functions. In Section 3 we introduce the elliptic differential operators and the corresponding Dirichlet-to-Neumann map that is used for the analysis of the algebraic eigenspaces. Here we treat second-order divergence form elliptic operators with (complex) L ∞ -coefficients of the form on bounded Lipschitz domains with non-local Robin boundary conditions. In this general situation it is necessary to pay special attention to the definition and properties of the co-normal and adjoint co-normal derivative, and to the properties of the corresponding sesquilinear forms and operators. Furthermore, the unique solvability of the homogeneous and inhomogeneous Dirichlet boundary value problems is discussed. For the convenience of the reader we provide proofs of these preparatory results in Section 3. Our main result on the characterization of Jordan chains of second-order elliptic partial differential operators with local or non-local Robin boundary conditions via Jordan chains of the Dirichletto-Neumann map λ → D(λ) and the boundary operator B is formulated and proved in Section 4. The proof is technical and requires the preparatory Lemma 4.5. Finally, in Subsection 5.1 we discuss a more regular situation in which the bounded domain Ω is assumed to have a C 2 -smooth boundary and the coefficients of the elliptic operator are slightly more regular. In this setting one then obtains a Dirichlet-to-Neumann operator acting from H 3/2 (Γ) into H 1/2 (Γ) and a variant of Theorem 4.1 for H 2 (Ω)-smooth Jordan chains. In Subsection 5.2 we reconsider the Dirichlet-to-Neumann operator on a Lipschitz domain, but now we treat the Dirichlet-to-Neumann operator acting from H 1 (Γ) into L 2 (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 and the 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 and part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

Jordan chains of operators and holomorphic operator functions
Throughout this paper the field is the complex numbers. Let A be a linear operator in a Banach space H. Further, let k ∈ N 0 , f 0 , . . . , f k ∈ H and λ 0 ∈ C. Then we say that the for all j ∈ {0, . . . , k} with f 0 = 0 and we set f −1 = 0. The vector f 0 is called an eigenvector of A at the eigenvalue λ 0 and the vectors f 1 , . . . , f k are said to be generalized eigenvectors of A at λ 0 . Note that the generalized eigenvectors are all nonzero. The notion of Jordan chains exists also for holomorphic operator functions and goes back to the work of M.V. Keldysh [Kel], for more details we also refer the reader to the monograph [Mark,§11]. Let H 1 and H 2 be Banach spaces, O ⊂ C an open set and for all λ ∈ O let M (λ) ∈ L(H 1 , H 2 ). 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 0 and ϕ 0 , . . . , ϕ k ∈ H 1 . Then we say that the vectors {ϕ 0 , . . . , ϕ k } form a Jordan chain for the function M (·) at λ 0 ∈ O if j l=0 1 l! M (l) (λ 0 )ϕ j−l = 0 for all j ∈ {0, . . . , k} and ϕ 0 = 0. The vector ϕ 0 is called an eigenvector of the operator function M (·) at the eigenvalue λ 0 and the vectors ϕ 1 , . . . , ϕ k are said to be generalized eigenvectors of M (·) at λ 0 .
Observe that in the special case H 1 = H 2 and C ∈ L(H 1 ) the notion of Jordan chain for the operator C at λ 0 ∈ C and the notion of Jordan chain for the function λ → C − λ at λ 0 ∈ C coincide.
For all k, l ∈ {1, . . . , d} fix c kl , b k , c k , c 0 ∈ L ∞ (Ω). We recall that the field is the complex numbers, so we emphasise that all coefficients are complex valued. Assume that there exists a µ > 0 such that The form a is continuous in the sense that there exists an M ≥ 0 such that |a(f, g)| ≤ M f H 1 (Ω) g H 1 (Ω) for all f, g ∈ H 1 (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 a closed sectorial form. Introduce In order to introduce the co-normal derivative we need a lemma. Note that the ellipticity condition on the principal coefficients is not needed in the next lemma.
for all g ∈ H 1 (Ω). We call γ N f the co-normal derivative of f . Denote by a D the restriction of a to H 1 0 (Ω)×H 1 0 (Ω). Then a D is a continous elliptic form and hence a closed sectorial form (cf. [AE3] Lemma 3.1.) Denote by A D the m-sectorial operator associated with the form a D . It follows that A D is the Dirichlet realization of A in L 2 (Ω) given by . Then the following assertions hold.
(b) For all ϕ ∈ H 1/2 (Γ) and all h ∈ L 2 (Ω) there exists a unique solution f ∈ H 1 (Ω) of the inhomogeneous boundary value problem Proof. '(a)'. The existence follows as in the proof of [AE6] Lemma 2.1. For completeness we give the details. There exists a T ∈ L(H 1 0 (Ω)) such that for all f, g ∈ H 1 0 (Ω). Further there exists an ω > 0 such that the sesquilinear form b : There exists an f 0 ∈ H 1 (Ω) such that Tr f 0 = ϕ. Hence there exists an h ∈ H 1 0 (Ω) such The uniqueness is easy. The continuity of the map follows from the closed graph theorem. '(b)'. By Statement (a) there exists an f 0 ∈ H 1 (Ω) such that (A − λ)f 0 = 0 and Tr f 0 = ϕ. Then f 0 + (A D − λ) −1 h is a solution to the problem (3.2). Again the uniqueness is easy.
Proof. '(a)'. Fix λ 0 ∈ ρ(A D ). By Lemma 3.2(a) there exists a unique g λ 0 ∈ H 1 (Ω) such that (A − λ 0 )g λ 0 = 0 and Tr g λ 0 = ϕ. Let λ ∈ ρ(A D ) and consider Then (A − λ)g = (A − λ)g λ 0 + (λ − λ 0 )g λ 0 = 0 and Tr g = Tr g λ 0 = ϕ. Since the solution of the homogeneous boundary value problem (A − λ)f = 0 with Tr f = ϕ, is unique by Lemma 3.2(a) it follows that g = g λ . Now the holomorphy of the resolvent λ → it follows that λ → D(λ) is holomorphic with respect to the weak operator topology on L(H 1/2 (Γ), H −1/2 (Γ)), and therefore it is also holomorphic with respect to the uniform operator topology. For all l ∈ N we denote the l-th derivative of λ → D(λ) at λ ∈ ρ(A D ) by D (l) (λ). Then according to Lemma 3.3(b) one has The dual form a * of a is defined by dom (a * ) = H 1 (Ω) and a * (f, g) = a(g, f ) for all f, g ∈ H 1 (Ω). So Obviously a * is of the same type as a, with c kl replaced by c lk , etc. Similar to the definition of A with respect to a, we can define the operator A : H 1 (Ω) → (H 1 0 (Ω)) * by As in Lemma 3.1 it follows that for all f ∈ H 1 (Ω) with Af ∈ L 2 (Ω), there exists a unique for all g ∈ H 1 (Ω). Using all definitions it is easy to prove the following version of Green's second identity.
So γ N f = BTr f and hence f ∈ dom A B . The converse inclusion follows similarly.

Jordan chains of Robin realizations
Adopt the assumptions and notation as in Section 3. In this section we formulate and prove our main result on the characterization of Jordan chains of the m-sectorial Robin realization A B of A via the operator function λ → D(λ) − B. Our goal is to show the following theorem.  Then {f 0 , . . . , f k } is a Jordan chain for A B at λ 0 .
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 following holds.  Remark 4.4. We can mention here that the assumption λ 0 ∈ ρ(A D ) in Theorem 4.1 and Corollary 4.2 is really needed. In fact, one may define the Dirichlet-to-Neumann graph as a linear relation consisting of the Cauchy data for all λ 0 ∈ σ p (A D ). By [Fil] Theorem 1 there exist µ > 0, λ ∈ R, u ∈ C ∞ c (R 3 ) \ {0} and a Hölder continuous function g : R 3 → [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 = 0 for all k, l ∈ {1, . . . , d} and f 0 = u| Ω . Let B ∈ L(L 2 (Γ)). Then f 0 is an eigenfunction of A B at λ. But Tr f 0 = 0. So one cannot drop the assumption λ 0 ∈ ρ(A D ) in Corollary 4.2(a).
Observe that the homogenenous and inhomogeneous boundary value problems in Theorem 4.1(b) and Corollary 4.2(b) admit unique solutions by Lemma 3.2. The proof of Theorem 4.1 requires quite some preparation. The next lemma is particularly useful; its proof is partly based on an argument that was given by V.A. Derkach for symmetric and selfadjoint linear relations in Krein spaces; see also [DM3] Section 7.4.4.
Remark 4.6. In the abstract setting of boundary triplets and their Weyl functions for adjoint pairs [LS, MM, Vai] it is known under a natural unique continuation hypothesis that the 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 the context of indefinite inner product spaces.
There are two other Dirichlet-to-Neumann operators that we consider in this section.

C 2 -domains
Throughout this subsection we suppose that Ω is a C 2 -domain, c kl ∈ C 1 (Ω) and b k = 0 for all k, l ∈ {1, . . . , d}. We summarise some regularity results that we need in this subsection.
The uniqueness is easy. The continuity follows from Lemma 3.2(a) and the closed graph theorem.
Next we consider holomorphy.
The alluded variation of Theorem 4.1 is as follows.
The proof is similar to the proof of Theorem 4.1, with obvious changes.
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 the restriction on the principal part of the elliptic operator. We collect the main properties of the operator D(λ) in the next proposition. (c) The map λ → D(λ) from ρ(A D ) into L(H 1 (Γ), L 2 (Γ)) is holomorphic.
Then argue as in the proof of Lemma 5.2. Now we are able to formulate another version of Theorem 4.1.
The proof is similar to the proof of Theorem 4.1, with obvious changes.