Boundary integral formulations of eigenvalue problems for elliptic differential operators with singular interactions and their numerical approximation by boundary element methods
BBOUNDARY INTEGRAL FORMULATIONS OF EIGENVALUEPROBLEMS FOR ELLIPTIC DIFFERENTIAL OPERATORSWITH SINGULAR INTERACTIONS AND THEIR NUMERICALAPPROXIMATION BY BOUNDARY ELEMENT METHODS
MARKUS HOLZMANN AND GERHARD UNGER
Abstract.
In this paper the discrete eigenvalues of elliptic second order dif-ferential operators in L p R n q , n P N , with singular δ - and δ -interactions arestudied. We show the self-adjointness of these operators and derive equivalentformulations for the eigenvalue problems involving boundary integral opera-tors. These formulations are suitable for the numerical computations of thediscrete eigenvalues and the corresponding eigenfunctions by boundary elementmethods. We provide convergence results and show numerical examples. Introduction
Schr¨odinger operators with singular interactions supported on sets of measurezero play an important role in mathematical physics. The simplest example areSchr¨odinger operators with point interactions, which were already introduced in thebeginnings of quantum mechanics [27, 35]. The importance of these models comesfrom the fact that they reflect the physical reality still to a reasonable exactnessand that they are explicitly solvable. The point interactions are used as idealizedreplacements for regular potentials, which are strongly localized close to those pointssupporting the interactions, and the eigenvalues can be computed explicitly via analgebraic equation involving the values of the fundamental solution corresponding tothe unperturbed operator evaluated at the interaction support, cf. the monograph[1] and the references therein.Inspired by this idea, Schr¨odinger operators with singular δ - and δ -interactionssupported on hypersurfaces (i.e. manifolds of codimension one like curves in R orsurfaces in R ) where introduced. Such interactions are used as idealized replace-ments of regular potentials which are strongly localized in neighborhoods of thesehypersurfaces e.g. in the mathematical analysis of leaky quantum graphs, cf. thereview [15] and the references therein, and in the theory of photonic crystals [18].Note that in the case of δ -potentials this idealized replacement is rigorously justifiedby an approximation procedure [3]. The self-adjointness and qualitative spectralproperties of Schr¨odinger operators with δ - and δ -interactions are well understood,see e.g. [6,7,11,15,16,29] and the references therein, and the discrete eigenvalues canbe characterized via an abstract version of the Birman Schwinger principle. How-ever, following the strategy from the point interaction model one arrives, instead of Key words and phrases. elliptic differential operators, δ and δ -interaction, discrete eigenvalues,integral operators, boundary element method. a r X i v : . [ m a t h . SP ] J u l M. HOLZMANN AND G. UNGER an algebraic equation, at a boundary integral equation involving the fundamentalsolution for the unperturbed operator.In this paper we suggest boundary element methods for the numerical approx-imations of these boundary integral equations. With this idea of computing theeigenvalues of the differential operators with singular interactions numerically, wegive a link of these models to the original explicitly solvable models with point in-teractions. As theoretical framework for the description of the eigenvalues in termsof boundary integral equations we use the theory of eigenvalue problems for holo-morphic and meromorphic Fredholm operator-valued functions [19, 20, 26]. For theapproximation of this kind of eigenvalue problems by the Galerkin method thereexists a complete convergence analysis in the case that the operator-valued functionis holomorphic [21,22,34]. This analysis provides error estimates for the eigenvaluesand eigenfunctions as well as results which guarantee that the approximation of theeigenvalue problems does not have so-called spurious eigenvalues, i. e., additionaleigenvalues which are not related to the original problem.Other approaches for the numerical approximation of eigenvalues of differentialoperators with singular interactions are based on finite element methods, where R n is replaced by a big ball, whose size can be estimated with the help of Agmon typeestimates. Moreover, in [12, 17, 30] it is shown in various settings in space dimen-sions n P t , u that Schr¨odinger operators with δ -potentials supported on curves(for n “
2) or surfaces (for n “
3) can be approximated in the strong resolventsense by Hamiltonians with point interactions. An improvement of this approachis presented in [14]. This allows also to compute numerically the eigenvalues of thelimit operator.Let us introduce our problem setting and give an overview of the main results.Consider a strongly elliptic and formally symmetric partial differential operator in R n , n P N , of the form P : “ ´ n ÿ j,k “ B k a jk B j ` n ÿ j “ ` a j B j ´ B j a j ˘ ` a, see Section 3 for details. Moreover, let Ω i be a bounded Lipschitz domain withboundary Σ : “ B Ω i , let Ω e : “ R n z Ω i , and let ν be the unit normal to Ω i . Eventually,let γ be the Dirichlet trace and B ν the conormal derivative at Σ (see (3.4) for thedefinition). We are interested in the eigenvalues of two kinds of perturbations of P as self-adjoint operators in L p R n q which are formally given by A α : “ P ` αδ Σ and B β : “ P ` β x δ Σ , ¨y δ Σ , where δ Σ is the Dirac δ -distribution supported on Σ and the interaction strengths α, β are real valued functions defined on Σ with α, β ´ P L p Σ q . For P “ ´ ∆ theseoperators have been intensively studied e.g. in [7, 11, 15, 16], for certain stronglyelliptic operators and smooth surfaces several properties of A α and B β have beeninvestigated in [6,29]. For the realization of A α as an operator in L p R n q we remarkthat if the distribution A α f is generated by an L -function, then f i/e : “ f æ Ω i/e has to fulfill(1.1) γf i “ γf e and B ν f e ´ B ν f i “ αγf on Σ , as then the singularities at Σ compensate, cf. [7]. In a similar manner, if thedistribution B β f is generated by an L -function, then f has to fulfill(1.2) B ν f i “ B ν f e and γf e ´ γf i “ β B ν f on Σ . Hence, the relations (1.1) and (1.2) are necessary conditions for a function f tobelong to the domain of definition of A α and B β , respectively. Our aims are toshow the self-adjointness of A α and B β in L p R n q and to fully characterize theirdiscrete spectra in terms of boundary integral operators. We pay particular atten-tion to establish formulations which fit in the framework of eigenvalue problems forholomorphic and meromorphic Fredholm operator-valued functions and which areaccessible for boundary element methods. This requires a thorough analysis of theinvolved boundary integral operators.When using boundary element methods for the approximations of discrete eigen-values of A α and B β it is convenient to consider the related transmission problems.A value λ belongs to the point spectrum of A α if and only if there exists a nontrivial f P L p R n q satisfying p P ´ λ q f “ R n z Σ ,γf i “ γf e , on Σ , B ν f e ´ B ν f i “ αγf on Σ . (1.3)Similarly, λ belongs to the point spectrum of B β if and only if there exists a non-trivial f P L p R n q satisfying p P ´ λ q f “ R n z Σ , B ν f i “ B ν f e on Σ ,γf e ´ γf i “ β B ν f on Σ . (1.4)This shows that the eigenvalue problems for A α and B β are closely related totransmission problems for P ´ λ , as they were treated in [24,25], and the strategiespresented there are useful for the numerical calculation of the eigenvalues of A α and B β .For the analysis of the spectra of A α and B β a good understanding of the unper-turbed operator A being the self-adjoint realization of P with no jump conditionat Σ and some operators related to the fundamental solution of P ´ λ are necessary.Assume for λ P ρ p A q Y σ disc p A q that G p λ ; x, y q is the integral kernel of a suitableparamatrix associated to P ´ λ which is explained in detail in Section 3; for λ inthe resolvent set ρ p A q it is in fact a fundamental solution for P ´ λ . We remarkthat the knowledge of G p λ ; x, y q or at least a good approximation of this function isessential for our numerical considerations. We introduce the single layer potentialSL p λ q and the double layer potential DL p λ q acting on sufficiently smooth functions ϕ : Σ Ñ C and x P R n z Σ asSL p λ q ϕ p x q : “ ż Σ G p λ ; x, y q ϕ p y q d σ p y q and DL p λ q ϕ p x q : “ ż Σ p B ν,y G p λ ; x, y qq ϕ p y q d σ p y q . As we will see, all solutions of p P ´ λ q f “ p λ q and DL p λ q . Moreover, the boundary integral operators which are formally M. HOLZMANN AND G. UNGER given by S p λ q ϕ : “ γ SL p λ q ϕ, T p λ q ϕ : “ B ν p SL p λ q ϕ q i ` B ν p SL p λ q ϕ q e , and T p λ q ϕ : “ γ p DL p λ q ϕ q i ` γ p DL p λ q ϕ q e , R p λ q ϕ : “ ´ B ν DL p λ q ϕ, will play an important role. While the properties of the above operators are well-known for many special cases, e.g. for P “ ´ ∆, the corresponding results are, tothe best of the authors’ knowledge, not easily accessible in the literature for general P . Hence, for completeness we spend some efforts in Section 3.3 to provide thoseproperties of the above integral operators which are needed for our considerations.Eventually, following a strategy from [9], we show that the discrete eigenvaluesof A can be characterized as the poles of an operator-valued function which isbuilt up by the operators S p λ q , T p λ q , T p λ q , and R p λ q ; see also [13] for relatedresults. Compared to [9] our formulation is particularly useful for the applicationof boundary element methods to compute the discrete eigenvalues of A numerically,as the appearing operators are easily accessible for numerical computations.In order to introduce A α and B β rigorously, consider the Sobolev spaces H P p Ω q : “ t f P H p Ω q : P f P L p Ω qu . Inspired by (1.1) and (1.2) we define A α as the partial differential operator in L p R n q given by A α f : “ P f i ‘ P f e , dom A α : “ (cid:32) f “ f i ‘ f e P H P p Ω i q ‘ H P p Ω e q : γf i “ γf e , B ν f e ´ B ν f i “ αγf ( , (1.5)and B β by B β f : “ P f i ‘ P f e , dom B β : “ (cid:32) f “ f i ‘ f e P H P p Ω i q ‘ H P p Ω e q : B ν f i “ B ν f e , γf e ´ γf i “ β B ν f ( . (1.6)In Section 4 and 5 we show the self-adjointness of these operators in L p R n q and viathe Weyl theorem that the essential spectra of A α and B β coincide with the essentialspectrum of the unperturbed operator A . Hence, to know the spectral profile of A α and B β we have to understand the discrete eigenvalues of these operators. Thecharacterization of the discrete eigenvalues of A α and B β in terms of boundaryintegral equations depends on the discrete spectrum of the unperturbed operator A being empty or not. Let us consider the first case. It turns out that λ P ρ p A q is a discrete eigenvalue of A α if and only if there exists a nontrivial ϕ P L p Σ q suchthat(1.7) p I ` α S p λ qq ϕ “ . Similarly, the existence of a discrete eigenvalue λ P ρ p A q of B β is equivalent to theexistence of a corresponding nontrivial ψ P H { p Σ q which satisfies(1.8) p β ´ ` R p λ qq ψ “ . As shown in Sections 4 and 5 the boundary integral formulations in (1.7) and (1.8)are eigenvalue problems for holomorphic Fredholm operator-valued functions. Theseeigenvalue problems can be approximated by standard boundary element methods.
The convergence of the approximations follows from well-known abstract conver-gence results [21, 22, 34], which are summarized in Section 2. In the case that σ disc p A q is not empty, still all eigenvalues of A α and B β in ρ p A q can be character-ized and computed using (1.7) and (1.8), respectively. For the possible eigenvalues A α and B β which lie in σ disc p A q also boundary integral formulations are providedwhich are accessible by boundary element methods and discussed in detail in Sec-tion 4 and 5.Finally, let us note that our model also contains certain classes of magneticSchr¨odinger operators with singular interactions with rather strong limitations forthe magnetic field. Nevertheless, one could use our strategy and the Birman-Schwinger principle for magnetic Schr¨odinger operators with more general magneticfields provided in [4, 30] to compute the discrete eigenvalues of such Hamiltoniansnumerically. Also, an extension of our results to Dirac operators with δ -shell inter-actions [5] would be of interest, but this seems to be a rather challenging problem.Let us shortly describe the structure of the paper. In Section 2 we recall somebasic facts on eigenvalue problems of holomorphic Fredholm operator-valued func-tions and on the approximation of this kind of eigenvalue problems by the Galerkinmethod. In Section 3 we introduce the elliptic differential operator P and the asso-ciated integral operators and investigate the properties of the unperturbed operator A . Sections 4 and 5 are devoted to the analysis of A α and B β , respectively. Weintroduce these operators as partial differential operators in L p R n q , show theirself-adjointness and derive boundary integral formulations to characterize their dis-crete eigenvalues. Moreover, we discuss how these boundary integral equations canbe solved numerically by boundary element methods, provide convergence results,and give some numerical examples. Notations.
Let X and Y be complex Hilbert spaces. The set of all anti-linearbounded functionals on X and Y are denoted by X ˚ and Y ˚ , respectively, and thesesquilinear duality product in X ˚ ˆ X , which is linear in the first and anti-linearin the second argument, is p¨ , ¨q ; the underlying spaces of the duality product willbe clear from the context. Next, the set of all bounded and everywhere definedlinear operators from X to Y is B p X, Y q ; if X “ Y , then we simply write B p X q : “ B p X, X q . For A P B p X, Y q the adjoint A ˚ P B p Y ˚ , X ˚ q is uniquely determined bythe relation p Ax, y q “ p x, A ˚ y q for all x P X and y P Y ˚ .If A is a self-adjoint operator in a Hilbert space, then its domain, range, andkernel are denoted by dom A , ran A , and ker A . The resolvent set, spectrum, dis-crete, essential, and point spectrum are ρ p A q , σ p A q , σ disc p A q , σ ess p A q , and σ p p A q ,respectively. Finally, if Λ is an open subset of C and A : Λ Ñ B p X, X ˚ q , then wesay that λ P Λ is an eigenvalue of A p¨q , if ker A p λ q ‰ t u . Acknowledgements.
We are specially grateful to O. Steinbach for encouragingus to work on this project. Moreover, we thank J. Behrndt and J. Rohleder forhelpful discussions and literature hints.
M. HOLZMANN AND G. UNGER Galerkin approximation of eigenvalue problems for holomorphicFredholm operator-valued functions
In this section we present basic results of the theory of eigenvalue problemsfor holomorphic Fredholm operator-valued functions [19, 26] and summarize mainresults of the convergence analysis of the Galerkin approximation of such eigenvalueproblems [21, 22, 36]. These results build the abstract framework which we willutilize in order to show the convergence of the boundary element method for theapproximation of the discrete eigenvalues of A α as well as of B β which lie in ρ p A q .Under specified conditions the convergence for discrete eigenvalues of A α and B β in σ disc p A q is also guaranteed.Let X be a Hilbert space and let Λ Ă C be an open and connected subsetof C . We consider an operator-valued function F : Λ Ñ B p X, X ˚ q which de-pends holomorphically on λ P Λ, i.e., for each λ P Λ the derivative F p λ q : “ lim λ Ñ λ λ ´ λ } F p λ q ´ F p λ q} L p X,X ˚ q exists. Moreover, we assume that F p λ q isa Fredholm operator of index zero for all λ P Λ and that it satisfies a so-calledG˚arding’s inequality, i. e., there exists a compact operator C p λ q : X Ñ X ˚ and aconstant c p λ q ą λ P Λ such that(2.1) | pp F p λ q ` C p λ qq u, u q | ě c p λ q} u } X for all u P X. We consider the nonlinear eigenvalue problem for the operator-valued func-tion F p¨q of the form: find eigenvalues λ P Λ and corresponding eigenelements u P X zt u such that(2.2) F p λ q u “ . In the following we assume that the set t λ P Λ : D F p λ q ´ P B p X ˚ , X qu is not empty.Then the set of eigenvalues in Λ has no accumulation points inside of Λ [19, Cor.XI 8.4]. The dimension of the null space ker F p λ q of an eigenvalue λ is called thegeometric multiplicity of λ . An ordered collection of elements u , u , . . . , u m ´ in X is called a Jordan chain of p λ, u q , if it is an eigenpair and if n ÿ j “ j ! F p j q p λ q u n ´ j “ n “ , , . . . , m ´ F p j q denotes the j th derivative. The length of any Jordan chainof an eigenvalue is finite [26, Lem. A.8.3]. Elements of any Jordan chain of aneigenvalue λ are called generalized eigenelements of λ . The closed linear space ofall generalized eigenelements of an eigenvalue λ is called generalized eigenspace of λ and is denoted by G p F , λ q . The dimension of the generalized eigenspace G p F , λ q is finite [26, Prop. A.8.4] and it is referred to as algebraic multiplicity of λ .2.1. Galerkin approximation.
For the approximation of the eigenvalue prob-lem (2.2) we consider a conforming Galerkin approximation. We assume that p X N q N P N is a sequence of finite-dimensional subspaces of X such that the orthog-onal projection P N : X Ñ X N converges pointwise to the identity I : X Ñ X , i.e.,for all u P X we have(2.3) } P N u ´ u } X “ inf v N P X N } v N ´ u } X Ñ N Ñ 8 . The Galerkin approximation of the eigenvalue problem reads as: find eigenpairs p λ N , u N q P Λ ˆ X N zt u such that(2.4) p F p λ N q u N , v N q “ v n P X N . For the formulation of the convergence results we need the definition of the gap δ V p V , V q of two subspaces V , V of a normed space V : δ V p V , V q : “ sup v P V } v } V “ inf v P V } v ´ v } V . Theorem 2.1.
Let F : Λ Ñ B p X, X ˚ q be a holomorphic operator-valued functionand assume that for each λ P Λ there exist a compact operator C p λ q : X Ñ X ˚ and a constant c p λ q ą such that inequality (2.1) is satisfied. Further, supposethat p X N q N P N is a sequence of finite-dimensional subspaces of X which fulfills theproperty (2.3) . Then the following holds true: (i) (Completeness of the spectrum of the Galerkin eigenvalue problem) For eacheigenvalue λ P Λ of the operator-valued function F p¨q there exists a sequence p λ N q N of eigenvalues of the Galerkin eigenvalue problem (2.4) such that λ N Ñ λ as N Ñ 8 . (ii) (Non-pollution of the spectrum of the Galerkin eigenvalue problem) Let K Ă Λ be a compact and connected set such that B K is a simple rectifiable curve.Suppose that there is no eigenvalue of F p¨q in K . Then there exists an N P N such that for all N ě N the Galerkin eigenvalue problem (2.4) hasno eigenvalues in K . (iii) Let D Ă Λ be a compact and connected set such that B D is a simple recti-fiable curve. Suppose that λ P ˚ D is the only eigenvalue of F in D . Thenthere exist an N P N and a constant c ą such that for all N ě N wehave: (a) For all eigenvalues λ N of the Galerkin eigenvalue problem (2.4) in D | λ ´ λ N | ď cδ X p G p F , λ q , X N q { (cid:96) δ X p G p F ˚ , λ q , X N q { (cid:96) holds, where F ˚ p¨q : “ p F p¨qq ˚ is the adjoint function with respect tothe pairing p¨ , ¨q for X ˚ ˆ X and (cid:96) is the maximal length of a Jordanchain corresponding to λ . (b) If p λ N , u N q is an eigenpair of (2.4) with λ N P D and } u N } X “ , then inf u P ker p F ,λ q } u ´ u N } X ď c p| λ N ´ λ | ` δ X p ker p F , λ q , X N qq . Proof.
The Galerkin method fulfills the required properties in order to apply theabstract convergence results in [21, 22, 36] to eigenvalue problems for holomorphicoperator-valued functions which satisfy inequality (2.1), see [34, Lem. 4.1]. We referto [21, Thm. 2] for assertion (i) and (ii), and to [22, Thm. 3] for (iii)a). The errorestimate in (iii)b) is a consequence of [36, Thm. 4.3.7]. (cid:3) Strongly elliptic differential operators and associated integraloperators
In this section we introduce the class of elliptic differential operators which willbe perturbed by the singular δ - and δ -interactions supported on a hypersurface M. HOLZMANN AND G. UNGER
Σ, and we introduce the integral operators S p λ q , T p λ q , T p λ q , and R p λ q in Sec-tion 3.3 in a mathematically rigorous way and recall their properties, which willbe of importance for our further studies. Eventually, in Section 3.4 we show howthe discrete eigenvalues of A can be characterized with the help of these boundaryintegral operators. But first, we introduce our notations for function spaces whichwe use in this paper.3.1. Function spaces.
For an open set Ω Ă R n , n P N , and k P N Y t8u we write C k p Ω q for the set of all k -times continuously differentiable functions and C b p Ω q : “ t f P C p Ω q : f, ∇ f are bounded u . Moreover, the Sobolev spaces of order s P R are denoted by H s p Ω q , see [28, Chap-ter 3] for their definition.In the following we assume that Ω Ă R n is a Lipschitz domain in the senseof [28, Definition 3.28]. We emphasize that Ω can be bounded or unbounded, but B Ω has to be compact. Note that in this case we can identify H s p R n zB Ω q with H s p Ω q ‘ H s p R n z Ω q . With the help of the integral on B Ω with respect to theHausdorff measure we get in a natural way the definition of L pB Ω q . In a similarflavor, we denote the Sobolev spaces on B Ω of order s P r , s by H s pB Ω q , see [28]for details on their definition. For s P r´ , s we define H s pB Ω q : “ p H ´ s pB Ω qq ˚ asthe anti-dual space of H ´ s pB Ω q .Finally, we recall that the Dirichlet trace operator C p Ω q Q f ÞÑ f | B Ω can beextended for any s P p , q to a bounded and surjective operator(3.1) γ : H s p Ω q Ñ H s ´ { pB Ω q ;cf. [28, Theorem 3.38].3.2. Strongly elliptic differential operators.
Let a jk , a j , a P C b p R n q , n P N ,and j, k P t , . . . , n u , and define the differential operator(3.2) P f : “ ´ n ÿ j,k “ B k p a jk B j f q ` n ÿ j “ ` a j B j f ´ B j p a j f q ˘ ` af in the sense of distributions. We assume that a jk “ a kj and that a is real valued;then P is formally symmetric. Moreover, we assume that P is strongly elliptic, thatmeans there exists a constant C ą x such that n ÿ j,k “ a jk p x q ξ j ξ k ě C | ξ | holds for all x P R n and all ξ P C n .Next, define for an open subset Ω Ă R n the sesquilinear form Φ Ω : H p Ω qˆ H p Ω q by(3.3) Φ Ω r f, g s : “ ż Ω « n ÿ j,k “ a jk B j f B k g ` n ÿ j “ ` a j pB j f q g ` f p a j B j g q ˘ ` af g ff d x. In the following assume that Ω Ă R n is a Lipschitz set, let ν be the unit normalvector field at B Ω pointing outwards Ω, denote by γ the Dirichlet trace operator, see (3.1), and introduce for f P H p Ω q the conormal derivative B ν f by(3.4) B ν f : “ n ÿ k “ ν k n ÿ j “ γ p a jk B j f q ` n ÿ j “ ν j γ p a j f q . Then one can show that(3.5) p P f, g q L p Ω q “ Φ Ω r f, g s ´ p B ν f, γg q L pB Ω q , f P H p Ω q , g P H p Ω q , holds. Next, we introduce the Sobolev space(3.6) H P p Ω q : “ (cid:32) f P H p Ω q : P f P L p Ω q ( , where P f is understood in the distributional sense. It is well known that theconormal derivative B ν has a bounded extension(3.7) B ν : H P p Ω q Ñ H ´ { pB Ω q , such that (3.5) extends to(3.8) p P f, g q L p Ω q “ Φ Ω r f, g s ´ p B ν f, γg q , f P H P p Ω q , g P H p Ω q , where the term on the boundary in (3.5) is replaced by the duality product in H ´ { p Σ q and H { p Σ q , see [28, Lemma 4.3]. We remark that this formula alsoholds for Ω “ R n , then the term on the boundary is not present.Our first goal is to construct the unperturbed self-adjoint operator A in L p R n q associated to P . With the help of [28, Theorem 4.7] it is not difficult to showthat the sesquilinear form Φ R n fulfills the assumptions of the first representationtheorem [23, Theorem VI 2.1], so we can define A as the self-adjoint operatorcorresponding to Φ R n . The following result is well-known, the simple proof is leftto the reader. Lemma 3.1.
Let P be given by (3.2) and let the form Φ R n be defined by (3.3) .Then Φ R n is densely defined, symmetric, bounded from below, and closed. Theself-adjoint operator A in L p R n q associated to Φ R n is (3.9) A f “ P f, dom A “ H p R n q . Assume that Ω i is a bounded Lipschitz domain in R n with boundary Σ : “ B Ω i ,let ν be the unit normal to Ω i , and set Ω e : “ R n z Ω i . Then it follows from [28,Theorem 4.20] that a function f “ f i ‘ f e P H P p Ω i q ‘ H P p Ω e q fulfills(3.10) f P dom A “ H p R n q ðñ γf i “ γf e and B ν f i “ B ν f e . Next, we review some properties of the resolvent of A which are needed later.In the following, let λ P ρ p A q Y σ disc p A q be fixed. Recall that a map G is called a paramatrix for P ´ λ in the sense of [28, Chapter 6], if there exist integral operators K , K with C -smooth integral kernels such that G p P ´ λ q u “ u ´ K u and p P ´ λ q G u “ u ´ K u holds for all u P E ˚ p R n q , where E ˚ p R n q is the set of all distributions with compactsupport, cf. [28]. A paramatrix is a fundamental solution for P ´ λ , if the aboveequation holds with K “ K “ p A ´ λ q by p P λ and set(3.11) P λ : “ I ´ p P λ . Note that P λ “ I for λ P ρ p A q and if t e , . . . e N u , N : “ dim ker p A ´ λ q , is a basisof ker p A ´ λ q for λ P σ disc p A q , then p P λ f “ N ÿ k “ p f, e k q L p R n q e k “ ż R n K p¨ , y q f p y q d y, K p x, y q : “ N ÿ k “ e k p x q e k p y q , for all f P L p R n q . We remark that the integral kernel K is a C -function byelliptic regularity [28, Theorem 4.20]. By the spectral theorem we have that A ´ λ is boundedly invertible in P λ p L p R n qq . Therefore, the map(3.12) G p λ q : “ P λ p A ´ λ q ´ P λ is bounded in L p R n q , and it is a paramatrix for P ´ λ , as(3.13) p P ´ λ q P λ p A ´ λ q ´ P λ f “ P λ p A ´ λ q ´ P λ p P ´ λ q f “ P λ f “ f ´ p P λ f holds for all f P C p R n q . Therefore, by [28, Theorem 6.3 and Corollary 6.5] thereexists an integral kernel G p λ ; x, y q such that for almost every x P R n (3.14) G p λ q f p x q “ ż R n G p λ ; x, y q f p y q d y, f P L p R n q . In the following proposition we show some additional mapping properties of G p λ q for λ P ρ p A q Y σ disc p A q ; they are standard and well-known, but for completenesswe give the proof of this proposition. Proposition 3.2.
Let A be defined by (3.9) , let λ P ρ p A qY σ disc p A q , and let G p λ q be given by (3.12) . Then, for any s P r´ , s the mapping G p λ q can be extended toa bounded operator (3.15) G p λ q : H s p R n q Ñ H s ` p R n q . Moreover, the map ρ p A q Q λ ÞÑ p A ´ λ q ´ is holomorphic in B p H s p R n q , H s ` p R n qq .Proof. Assume that λ P ρ p A q Y σ disc p A q is fixed. First, we show that(3.16) G p λ q : L p R n q Ñ H p R n q is bounded. The operator in (3.16) is well-defined, as ran G p λ q “ ran P λ p A ´ λ q ´ P λ “ P λ dom p A ´ λ q Ă H p R n q . Moreover, we claim that the operatorin (3.16) is closed, then it is also bounded by the closed graph theorem. Let p f n q Ă L p R n q be a sequence and let f P L p R n q and g P H p R n q be such that f n Ñ f in L p R n q and G p λ q f n Ñ g in H p R n q . Since G p λ q is bounded in L p R n q , we get G p λ q f n Ñ G p λ q f in L p R n q . Moreover, as H p R n q is continuously embedded in L p R n q , we also have G p λ q f n Ñ g in L p R n q . Hence, we conclude G p λ q f “ g , which shows that the operator in (3.16) is closedand thus, bounded.Since the operator in (3.16) is bounded for any λ P ρ p A qY σ disc p A q , we concludeby duality that also G p λ q : H ´ p R n q Ñ L p R n q is bounded. Therefore, interpolation yields that the mapping property (3.15) holdsalso for all s P p´ , q .In order to show that λ ÞÑ p A ´ λ q ´ is holomorphic in B p H s p R n q , H s ` p R n qq for any s P r´ , s in a fixed point λ P ρ p A q , we note that the resolvent identityimplies “ ´ p λ ´ λ qp A ´ λ q ´ ‰ p A ´ λ q ´ “ p A ´ λ q ´ . If λ is close to λ , we deduce from the Neumann formula that 1 ´p λ ´ λ qp A ´ λ q ´ is boundedly invertible in H s ` p R n q and hence, p A ´ λ q ´ “ “ ´ p λ ´ λ qp A ´ λ q ´ ‰ ´ p A ´ λ q ´ . In particular, p A ´ λ q ´ is uniformly bounded in B p H s p R n q , H s ` p R n qq for λ belonging to a small neighborhood of λ and continuous in λ . Employing this andonce more the resolvent identity p A ´ λ q ´ ´ p A ´ λ q ´ “ p λ ´ λ qp A ´ λ q ´ p A ´ λ q ´ , we find that ρ p A q Q λ ÞÑ p A ´ λ q ´ is holomorphic in B p H s p R n q , H s ` p R n qq . (cid:3) Surface potentials associated to P . In this section we introduce severalfamilies of integral operators associated to the paramatrix G p λ q which will be ofimportance in the study of A α and B β and for the numerical calculation of theireigenvalues. Remark that many of the properties shown below are well known forspecial realizations of P , for instance P “ ´ ∆, but for completeness we also providethe proofs for general P .Throughout this section assume that Σ is the boundary of a bounded Lipschitzdomain Ω i , set Ω e : “ R n z Ω i , and let ν be the unit normal to Ω i . If f is a functiondefined on R n , then in the following we will often use the notations f i : “ f æ Ω i and f e : “ f æ Ω e .Recall that the Dirichlet trace operator γ : H p R n q Ñ H { p Σ q is boundedby (3.1). Hence, it has a bounded dual γ ˚ : H ´ { p Σ q Ñ H ´ p R n q . This allows usto define for λ P ρ p A q Y σ disc p A q the single layer potential (3.17) SL p λ q : “ G p λ q γ ˚ : H ´ { p Σ q Ñ H p R n q . By the mapping properties of γ ˚ and Proposition 3.2 the map SL p λ q is well-definedand bounded. Moreover, we have ran SL p λ q Ă ran P λ “ L p R n q a ker p A ´ λ q .With the help of (3.14) and duality, it is not difficult to show that SL p λ q acts onfunctions ϕ P L p Σ q and almost every x P R n z Σ asSL p λ q ϕ p x q “ ż Σ G p λ ; x, y q ϕ p y q d σ p y q . Some further properties of SL p λ q are collected in the following lemma. In particular,the map SL p λ q plays an important role to construct eigenfunctions of the operator A α defined in (1.5). For that, we prove in the lemma below the correspondence ofthe range of SL p λ q with all solutions f P H P p R n z Σ q of the equation p P ´ λ q f “ R n z Σ and γf i “ γf e . For this purpose we define for λ P ρ p A q Y σ disc p A q the set(3.18) M λ : “ t ϕ P H ´ { p Σ q : p ϕ, γf q “ @ f P ker p A ´ λ qu . We remark that M λ “ H ´ { p Σ q for λ P ρ p A q . Lemma 3.3.
Let SL p λ q , λ P ρ p A q Y σ disc p A q , be defined by (3.17) . Then thefollowing is true: (i) We have ran SL p λ q Ă H P p R n z Σ q and (3.19) SL p λ qp M λ q ‘ ker p A ´ λ q “ (cid:32) f P H p R n q : p P ´ λ q f “ in R n z Σ ( . (ii) Let B ν be the conormal derivative defined by (3.8) . Then for any ϕ P H ´ { p Σ q the jump relations γ p SL p λ q ϕ q i ´ γ p SL p λ q ϕ q e “ and B ν p SL p λ q ϕ q i ´ B ν p SL p λ q ϕ q e “ ϕ hold. (iii) The map ρ p A q Q λ ÞÑ SL p λ q is holomorphic in B p H ´ { p Σ q , H P p R n z Σ qq .Proof. (i)–(ii) Let t e , . . . , e N u be a basis of ker p A ´ λ q (we use the conventionthat this set is empty for λ P ρ p A q ). Since G p λ q is a paramatrix for P ´ λ , theconsiderations in [28, equation (6.19)] and (3.13) imply for ϕ P H ´ { p Σ q that(3.20) p P ´ λ q SL p λ q ϕ “ ´ p P λ γ ˚ ϕ “ ´ N ÿ j “ p ϕ, γe j q e j on R n z Σ . This implies, in particular, that ran SL p λ q Ă H P p R n z Σ q and hence, B ν p SL p λ q ϕ q i { e is well-defined for ϕ P H ´ { p Σ q by (3.7). The jump relations in item (ii) are shownin [28, Theorem 6.11]. Furthermore, (3.20) implies p P ´ λ q SL p λ q ϕ “ R n z Σ for ϕ P M λ and thus,(3.21) SL p λ qp M λ q ‘ ker p A ´ λ q Ă (cid:32) f P H p R n q : p P ´ λ q f “ R n z Σ ( . Next, we verify the second inclusion in (3.19). Let f P H p R n q X H P p R n z Σ q such that p P ´ λ q f “ R n z Σ. Set ϕ : “ B ν f i ´ B ν f e P H ´ { p Σ q . We claimthat ϕ P M λ . For λ P ρ p A q this is clear by the definition of M λ in (3.18). For λ P σ disc p A q Ă R we get with (3.8) applied in Ω i and Ω e (note that ν is pointingoutside Ω i and inside Ω e ) for any g P ker p A ´ λ q Ă H p R n qp ϕ, γg q “ p B ν f i ´ B ν f e , γg q ´ p γf, B ν g i ´ B ν g e q“ p f, P g q L p R n q ´ p P f, g q L p R n q “ p f, λg q L p R n q ´ p λf, g q L p R n q “ , which implies ϕ P M λ . Next, consider the function h : “ f ´ SL p λ q ϕ . Then h P H p R n q and by (ii) we have B ν h i ´ B ν h e “ B ν f i ´ B ν f e ´ ` B ν p SL p λ q ϕ q i ´ B ν p SL p λ q ϕ q e ˘ “ ϕ ´ ϕ “ . Hence, (3.10) yields h P dom A . Eventually, due to the properties of f and SL p λ q ϕ for ϕ P M λ we conclude p A ´ λ q h “ p P ´ λ q h i ‘ p P ´ λ q h e “ p P ´ λ qp f i ´ p SL p λ q ϕ q i q ‘ p P ´ λ qp f e ´ SL p λ q ϕ q e q “ . This gives h “ f ´ SL p λ q ϕ P ker p A ´ λ q . Therefore, we have also verified(3.22) (cid:32) f P H p R n q : p P ´ λ q f “ R n z Σ ( Ă ran SL p λ q ‘ ker p A ´ λ q . The inclusions in (3.21) and (3.22) imply finally (3.19). (iii) By the definition of SL p λ q and Proposition 3.2 we have that SL p λ q is holomor-phic in B p H ´ { p Σ q , H p R n qq . Since P SL p λ q ϕ “ λ SL p λ q ϕ in R n z Σ for λ P ρ p A q by (i), we find that the H -norm is equivalent to the norm in H P p R n z Σ q onran SL p λ q . Therefore, SL p λ q is also holomorphic in B p H ´ { p Σ q , H P p R n z Σ qq . (cid:3) Two important objects associated to SL p λ q are the single layer boundary integraloperator S p λ q , which is defined by(3.23) S p λ q : H ´ { p Σ q Ñ H { p Σ q , S p λ q ϕ “ γ SL p λ q ϕ “ γ G p λ q γ ˚ ϕ, and the mapping T p λ q , which is given by(3.24) T p λ q : H ´ { p Σ q Ñ H ´ { p Σ q , T p λ q ϕ “ B ν p SL p λ q ϕ q i ` B ν p SL p λ q ϕ q e . The operators S p λ q and T p λ q have for a density ϕ P L p Σ q and almost all x P Σthe integral representations S p λ q ϕ p x q “ ż Σ G p λ ; x, y q ϕ p y q d σ p y q and T p λ q ϕ p x q “ ε Œ ż Σ z B p x,ε q B ν,x G p λ ; x, y q ϕ p y q d σ p y q . Some further properties of S p λ q and T p λ q are stated in the following lemma: Lemma 3.4.
Let S p λ q and T p λ q , λ P ρ p A q Y σ disc p A q , be defined by (3.23) and (3.24) , respectively. Then, the following is true: (i) The restriction S p λ q : “ S p λ q æ L p Σ q has the mapping property S p λ q : L p Σ q Ñ H p Σ q . In particular, S p λ q is compact in L p Σ q . (ii) S p λ q is a Fredholm operator with index zero and there exist a compact op-erator C p λ q : H ´ { p Σ q Ñ H { p Σ q and a constant c p λ q ą such that Re p ϕ, p S p λ q ` C p λ qq ϕ q ě c p λ q} ϕ } H ´ { p Σ q holds for all ϕ P H ´ { p Σ q . (iii) The maps ρ p A q Q λ ÞÑ S p λ q and ρ p A q Q λ ÞÑ T p λ q are holomorphic in B p H ´ { p Σ q , H { p Σ qq and B p H ´ { p Σ qq , respectively. (iv) For any ϕ P H ´ { p Σ q B ν p SL p λ q ϕ q i “ p ϕ ` T p λ q ϕ q and B ν p SL p λ q ϕ q e “ p´ ϕ ` T p λ q ϕ q hold.Proof. For the proof of the mapping property of S p λ q in (i) we refer to the discus-sion after [28, Theorem 6.12], the compactness of S p λ q follows then from the factthat H p Σ q is compactly embedded in L p Σ q . Statement (ii) is shown in [28, The-orem 7.6]. Item (iii) is a consequence of Lemma 3.3 (iii) and the mapping prop-erties of γ and B ν , respectively. Finally, statement (iv) follows immediately fromLemma 3.3 (ii) and the definition of T p λ q in (3.24). (cid:3) Next, we define the double layer potential associated to P ´ λ . For that werecall the definition of the conormal derivative B ν from (3.4) and note that B ν : H p R q Ñ L p Σ q is bounded. Hence, it admits a dual B ˚ ν P B p L p Σ q , H ´ p R n qq and with the help of Proposition 3.2 (applied for s “ ´
2) we can define the doublelayer potential as the bounded operator(3.25) DL p λ q : “ G p λ q B ˚ ν : L p Σ q Ñ L p R n q . Since ran G p λ q Ă L p R n q a ker p A ´ λ q , we have ran DL p λ q Ă L p R n q a ker p A ´ λ q .Using (3.14) and duality it is not difficult to show that DL p λ q acts on functions ϕ P L p Σ q and almost all x P R n z Σ asDL p λ q ϕ p x q “ ż Σ p B ν,y G p λ ; x, y qq ϕ p y q d σ p y q . Some further properties of DL p λ q are collected in the following lemma. In par-ticular, the map DL p λ q plays an important role to construct eigenfunctions of theoperator B β defined in (1.6). For that, we investigate the correspondence of therange of DL p λ q with all solutions f P H P p R n z Σ q of the equation p P ´ λ q f “ R n z Σ and B ν f i “ B ν f e . For this purpose we define for λ P ρ p A q Y σ disc p A q the set(3.26) N λ : “ t ϕ P H { p Σ q : p ϕ, B ν f q “ @ f P ker p A ´ λ qu . We remark that N λ “ H { p Σ q for λ P ρ p A q . In analogy to Lemma 3.3 we havethe following properties of DL p λ q . Lemma 3.5.
Let DL p λ q , λ P ρ p A q Y σ disc p A q , be defined by (3.25) . Then thefollowing is true: (i) The restriction of DL p λ q onto H { p Σ q gives rise to a bounded operator DL p λ q : H { p Σ q Ñ H P p R n z Σ q and DL p λ qp N λ q ‘ ker p A ´ λ q“ (cid:32) f P H P p R n z Σ q : B ν f i “ B ν f e , p P ´ λ q f “ in R n z Σ ( . (3.27)(ii) Let B ν be the conormal derivative defined by (3.8) . Then for any ϕ P H { p Σ q the jump relations γ p DL p λ q ϕ q e ´ γ p DL p λ q ϕ q i “ ϕ and B ν p DL p λ q ϕ q i ´ B ν p DL p λ q ϕ q e “ hold. (iii) The map ρ p A q Q λ ÞÑ DL p λ q is holomorphic in B p H { p Σ q , H P p R n z Σ qq .Proof. The proofs of many statements of this lemma are analogous to the ones inLemma 3.3, so we point out only the main differences. Since G p λ q is a paramatrix for P ´ λ , the considerations in [28, equation (6.19)] and (3.13) imply for ϕ P H { p Σ q that(3.28) p P ´ λ q DL p λ q ϕ “ ´ p P λ B ˚ ν ϕ on R n z Σ . In particular, P p DL ϕ q i { e P L p Ω i { e q . Next, we show that DL p λ q : H { p Σ q Ñ H P p R n z Σ q is bounded. Using the last observation and the closed graph theorem itis enough to verify(3.29) DL p λ q ϕ P H p R n z Σ q for ϕ P H { p Σ q ;cf. the proof of (3.16) for a similar argument. To prove (3.29) choose R ą i is contained in the open ball B p , R q of radius R centered at the originand a cutoff function χ P C p R n q which is supported in B p , R ` q and satisfies χ æ B p , R q ”
1. Moreover, let ϕ P H { p Σ q be fixed. Then χ DL p λ q ϕ P H p R n z Σ q by [28, Theorem 6.11]. Furthermore, p ´ χ q DL p λ q ϕ belongs to L p R n q and by theproduct rule we have P p ´ χ q DL p λ q ϕ “ p ´ χ q P DL p λ q ϕ ´ n ÿ j,k “ “ a jk pB k p ´ χ qqpB j DL p λ q ϕ q ` DL p λ q ϕ B k p a jk B j p ´ χ qq` a jk pB j p ´ χ qqpB k DL p λ q ϕ q ‰ ` DL p λ q ϕ n ÿ j “ r a j B j p ´ χ q ´ a j B j p ´ χ qs . Since supp ∇ p ´ χ q “ supp ∇ χ Ă B p , R ` q , we have again with the help of [28,Theorem 6.11] that pB k p ´ χ qqpB j DL p λ q ϕ q P L p R n q and thus with P DL p λ q ϕ P L p R n q and a j , a jk P C b p R n q we obtain P p ´ χ q DL p λ q ϕ P L p R q . Therefore,we conclude from elliptic regularity that p ´ χ q DL p λ q ϕ P H p R n q . This implieseventually that DL p λ q ϕ “ χ DL p λ q ϕ ` p ´ χ q DL p λ q ϕ P H p R n z Σ q and thus (3.29).Next, item (ii) is shown in [28, Theorem 6.11]. Furthermore, the relation (3.27)can be shown in the same way as (3.19) using (3.28) instead of (3.20).In order to prove statement (iii), let λ , λ P ρ p A q . Using the resolvent identitywe have DL p λ q ´ DL p λ q “ ` p A ´ λ q ´ ´ p A ´ λ q ´ ˘ B ˚ ν “ p λ ´ λ qp A ´ λ q ´ p A ´ λ q ´ B ˚ ν . (3.30)Since p A ´ λ q ´ p A ´ λ q ´ P B p H ´ p R n q , H p R n qq is continuous in λ in thistopology, see Proposition 3.2, we conclude that DL p λ q : H { p Σ q Ñ H P p R n z Σ q isholomorphic. (cid:3) Two important objects associated to DL p λ q are the hypersingular boundary in-tegral operator R p λ q , which is defined by(3.31) R p λ q : H { p Σ q Ñ H ´ { p Σ q , R p λ q ϕ “ ´ B ν DL p λ q ϕ “ ´ B ν G p λ q B ˚ ν ϕ, and the operator(3.32) T p λ q : H { p Σ q Ñ H { p Σ q , T p λ q ϕ “ γ p DL p λ q ϕ q i ` γ p DL p λ q ϕ q e . It follows from Lemma 3.5 (i) and (3.7) that R p λ q and T p λ q are well-defined andbounded. While T p λ q has for a continuous density ϕ P C p Σ q and almost all x P Σ a representation as a strongly singular integral operator, T p λ q ϕ p x q “ ε Œ ż Σ z B p x,ε q p B ν,y G p λ ; x, y qq ϕ p y q d σ p y q , the hypersingular operator R p λ q can be only written as finite part integral R p λ q ϕ p x q “ ´ f.p. ε Œ ż Σ z B p x,ε q B ν,x p B ν,y G p λ ; x, y qq ϕ p y q d σ p y q , see [28, Section 7] for details. However, for special realizations of P the dualityproduct p R p λ q ϕ, ψ q can be computed in a more convenient way, cf. e.g. [28, The-orem 8.21]. Some further properties of R p λ q and T p λ q are stated in the followinglemma: Lemma 3.6.
Let R p λ q and T p λ q , λ P ρ p A q Y σ disc p A q , be defined by (3.31) and (3.32) , respectively. Then, the following is true: (i) R p λ q is a Fredholm operator with index zero and there exist a compactoperator C p λ q : H { p Σ q Ñ H ´ { p Σ q and a constant c p λ q ą such that Re p ϕ, p R p λ q ` C p λ qq ϕ q ě c p λ q} ϕ } H { p Σ q holds for all ϕ P H { p Σ q . (ii) The maps ρ p A q Q λ ÞÑ R p λ q and ρ p A q Q λ ÞÑ T p λ q are holomorphic in B p H { p Σ q , H ´ { p Σ qq and B p H { p Σ qq , respectively. (iii) For any ϕ P H { p Σ q γ p DL p λ q ϕ q i “ p´ ϕ ` T p λ q ϕ q and γ p DL p λ q ϕ q e “ p ϕ ` T p λ q ϕ q hold. (iv) For all λ, ν P ρ p A q the difference T p λ q ´ T p ν q is compact. (v) The relation p ϕ, T p λ q ψ q “ p T p λ q ϕ, ψ q holds for all ϕ P H ´ { p Σ q and ψ P H { p Σ q .Proof. Item (i) follows immediately from [28, Theorem 7.8]. Assertion (ii) is aconsequence of Lemma 3.5 (iii) and the mapping properties of γ and B ν in (3.1)and (3.7). Next, the claim of item (iii) follows directly from Lemma 3.5 (ii) andthe definition of T p λ q .To show statement (iv) assume that λ ‰ ν P ρ p A q . As in (3.30) we see thatDL p λ q ´ DL p ν q : L p Σ q Ñ H p R n q is bounded. Since H p R n q is boundedly embed-ded in H p R n q , we deduce with the mapping properties of γ from (3.1) that T p λ q ´ T p ν q “ p ν ´ λ q γ p A ´ ν q ´ p A ´ λ q ´ B ˚ ν is bounded from L p Σ q to H { p Σ q . Since H { p Σ q is compactly embedded in L p Σ q ,we conclude eventually that T p λ q ´ T p ν q is compact in H { p Σ q .Finally, statement (v) is shown in [28, Chapter 7], since the operator T ˚ in [28,Chapter 7] coincides with T p λ q . (cid:3) Characterization of discrete eigenvalues of A . In this section we showhow the discrete eigenvalues of A can be characterized with the help of the bound-ary integral operators S p λ q , T p λ q , T p λ q , and R p λ q . For that purpose we followclosely considerations from [9], but we adapt the arguments to obtain a formula-tion on more general hypersurfaces Σ which is also more convenient for numericalconsiderations.We define for λ P ρ p A q the operator A p λ q : H ´ { p Σ q ˆ H { p Σ q Ñ H { p Σ q ˆ H ´ { p Σ q , A p λ q ˆ ϕψ ˙ “ ˆ γ ` SL p λ q ϕ ` DL p λ q ψ ˘ i ´ B ν ` SL p λ q ϕ ` DL p λ q ψ ˘ e ˙ . (3.33)Due to the mapping properties of γ from (3.1) and B ν from (3.7) we get withLemma 3.3 (i) and Lemma 3.5 (i) that A p λ q is well-defined and bounded. WithLemma 3.4 (iv) and Lemma 3.6 (iii) we see that A p λ q can be written as the blockoperator matrix(3.34) A p λ q “ ˆ S p λ q p´ I ` T p λ qq p I ´ T p λ q q R p λ q ˙ . Some basic properties of A p λ q are collected in the following lemma: Lemma 3.7.
Let A p λ q , λ P ρ p A q , be defined by (3.33) . Then the following is true: (i) The map ρ p A q Q λ ÞÑ A p λ q is holomorphic. (ii) There exists a compact operator K p λ q and a constant c p λ q ą such that ˇˇˇˇˆ p A p λ q ` K p λ qq ˆ ϕψ ˙ , ˆ ϕψ ˙˙ˇˇˇˇ ě c p λ q ` } ϕ } H ´ { p Σ q ` } ψ } H { p Σ q ˘ holds for all ϕ P H ´ { p Σ q and ψ P H { p Σ q , where the duality product isthe one for the pairing H { p Σ q ˆ H ´ { p Σ q and H ´ { p Σ q ˆ H { p Σ q .Proof. Assertion (i) follows from Lemma 3.4 (iii) and Lemma 3.6 (ii), as S p λ q , T p λ q , T p λ q , and R p λ q are holomorphic. To prove item (ii) we compute ˆ A p λ q ˆ ϕψ ˙ , ˆ ϕψ ˙ ˙ “ ˆˆ S p λ q p´ I ` T p λ qq p I ´ T p λ q q R p λ q ˙ ˆ ϕψ ˙ , ˆ ϕψ ˙˙ “ p S p λ q ϕ, ϕ q ` p R p λ q ψ, ψ q ` ` p ϕ, ψ q ´ p ψ, ϕ q ˘ ` ` p T p λ q ψ, ϕ q ´ p ϕ, T p λ q ψ q ˘ ` ` p ϕ, T p λ q ψ q ´ p T p λ q ϕ, ψ q ˘ . With Lemma 3.6 (v) we have ` ϕ, T p λ q ψ ˘ ´ ` T p λ q ϕ, ψ ˘ “ ` ϕ, p T p λ q ´ T p λ qq ψ ˘ and the operator T p λ q ´ T p λ q is compact by Lemma 3.6 (iv). Therefore, we getwith a compact operator K p λ q Re ˆ A p λ q ˆ ϕψ ˙ , ˆ ϕψ ˙ ˙ “ Re ` p S p λ q ϕ, ϕ q ` p R p λ q ψ, ψ q ` ` ϕ, p T p λ q ´ T p λ qq ψ ˘˘ ě c p λ q ` } ϕ } H ´ { p Σ q ` } ψ } H { p Σ q ˘ ` Re ˆ K p λ q ˆ ϕψ ˙ , ˆ ϕψ ˙˙ , which implies because of | z | ě Re z for z P C the claimed result. (cid:3) In the following theorem we characterize the discrete eigenvalues of A withthe help of the operator-valued function A . For that we define for a number λ P σ disc p A qY ρ p A q “ C z σ ess p A q , for which there exists an ε ą B p λ , ε qzt λ u Ă ρ p A q , the map(3.35) R A p λ q : “ lim λ Ñ λ p λ ´ λ q A p λ q . The proof of the following theorem follows closely ideas from [9, Theorem 3.2], butthe operator A p λ q appearing in our formulation is easier accessible for numericalapplications as the map M p λ q in [9] since it consists of explicitly computable integraloperators. Theorem 3.8.
A number λ belongs to the discrete spectrum of A if and only if λ is a pole of A p λ q . Moreover, (3.36) ran R A p λ q “ (cid:32) p γf, B ν f q J : f P ker p A ´ λ q ( holds.Proof. Let λ R σ ess p A q . It suffices to show that (3.36) is true. Let µ P C z R befixed and let p P λ be the orthogonal projection in L p R n q onto ker p A ´ λ q . Weclaim first that(3.37) ker p A ´ λ q “ (cid:32) p P λ r SL p µ q ϕ ` DL p µ q ψ s : ϕ P H ´ { p Σ q , ψ P H { p Σ qu . To show this assume that f P ker p A ´ λ q is such that0 “ ` f, p P λ r SL p µ q ϕ ` DL p µ q ψ s ˘ L p R n q “ ` f, SL p µ q ϕ ` DL p µ q ψ ˘ L p R n q holds for all ϕ P H { p Σ q and ψ P H ´ { p Σ q . Since f P ker p A ´ λ q , we have p A ´ µ q ´ f “ p λ ´ µ q ´ f and thus, the definitions of SL p µ q and DL p µ q lead to0 “ ` f, p A ´ µ q ´ γ ˚ ϕ ` p A ´ µ q ´ B ˚ ν ψ ˘ L p R n q “ ` γ p A ´ µ q ´ f, ϕ ˘ ` ` B ν p A ´ µ q ´ f, ψ ˘ “ λ ´ µ “ p γf, ϕ q ` p B ν f, ψ q ‰ . Since this is true for all ϕ P H { p Σ q and ψ P H ´ { p Σ q , we conclude that γf “ B ν f “
0. It follows from [8, Proposition 2.5] (this result and its proof are also truefor unbounded domains) that f “
0. Since for λ R σ ess p A q the set ker p A ´ λ q isfinite-dimensional, (3.37) is shown.We are now prepared to prove (3.36). By the spectral theorem the resolvent of A can be written in a small neighborhood of λ as p A ´ µ q ´ “ λ ´ µ p P λ ` F p µ q , where F p µ q is a locally bounded and continuous operator in µ . Hence, we concludethat R A p λ q can be a nontrivial operator, only if p P λ is nontrivial, and thatran R A p λ q Ă (cid:32) p γf, B ν f q J : f P ker p A ´ λ q ( . To show the other inclusion in (3.36), let f P ker p A ´ λ q , fix µ P C z R , and choose ϕ P H ´ { p Σ q and ψ P H { p Σ q such that f “ p P λ r SL p µ q ϕ ` DL p µ q ψ s ; such a choiceis always possible by (3.37). Note that according to the spectral theorem we have p P λ g “ lim λ Ñ λ p λ ´ λ qp A ´ λ q ´ g , where the limit is the one in L p R n q . Hence,we find ˆ γf B ν f ˙ “ ˆ γ B ν ˙ p A ´ µ q ´ p A ´ µ q p P λ r SL p µ q ϕ ` DL p µ q ψ s“ p λ ´ µ q ˆ γ B ν ˙ p A ´ µ q ´ p P λ r SL p µ q ϕ ` DL p µ q ψ s“ p λ ´ µ q ˆ γ B ν ˙ p A ´ µ q ´ lim λ Ñ λ p λ ´ λ qp A ´ λ q ´ r SL p µ q ϕ ` DL p µ q ψ s . Note that the mapping ˆ γ B ν ˙ p A ´ µ q ´ : L p R n q Ñ H { p Σ q ˆ H ´ { p Σ q is continuous. Hence, we conclude ˆ γf B ν f ˙ “ lim λ Ñ λ p λ ´ λ qp λ ´ µ q ˆ γ B ν ˙ p A ´ µ q ´ p A ´ λ q ´ r SL p µ q ϕ ` DL p µ q ψ s“ lim λ Ñ λ p λ ´ λ qp λ ´ µ q ˆ γ B ν ˙ p A ´ µ q ´ p A ´ λ q ´ p A ´ µ q ´ r γ ˚ ϕ ` B ˚ ν ψ s . Applying two times the resolvent identity, we find first for g P L p R n q that p A ´ µ q ´ p A ´ λ q ´ p A ´ µ q ´ g “ µ ´ λ rp A ´ µ q ´ ´ p A ´ λ q ´ sp A ´ µ q ´ g “ µ ´ λ p A ´ µ q ´ g ´ p µ ´ λ q rp A ´ µ q ´ ´ p A ´ λ q ´ s g. With a continuity argument this extends to all g P H ´ p R n q . Using this, we findfinally ˆ γf B ν f ˙ “ lim λ Ñ λ p λ ´ λ qp λ ´ µ q ˆ γ B ν ˙ p A ´ µ q ´ p A ´ λ q ´ p A ´ µ q ´ r γ ˚ ϕ ` B ˚ ν ψ s“ lim λ Ñ λ p λ ´ λ qp λ ´ µ qp λ ´ µ q ˆ γ B ν ˙ rp A ´ λ q ´ γ ˚ ϕ ` p A ´ λ q ´ B ˚ ν ψ s“ lim λ Ñ λ p λ ´ λ qp λ ´ µ qp λ ´ µ q A p λ q ˆ ϕψ ˙ “ λ ´ µ R A p λ q ˆ ϕψ ˙ , which shows that also the second inclusion in (3.36) is true. This finishes the proofof this theorem. (cid:3) Elliptic differential operators with δ -potentials supported oncompact Lipschitz smooth surfaces This section is devoted to the study of the spectral properties of the differentialoperator which is formally given by A α : “ P ` αδ Σ . First, we introduce A α in Sec-tion 4.1 as an operator in L p R n q and show its self-adjointness; in this procedure wealso obtain in Proposition 4.2 the Birman-Schwinger principle to characterize thediscrete eigenvalues of A α via boundary integral equations. Then, in Section 4.2 wediscuss how these boundary integral equations can be solved numerically by bound-ary element methods. Finally, in Section 4.3 we show some numerical examples. Definition and self-adjointness of A α . As usual, Ω i Ă R n is a boundedLipschitz domain with boundary Σ : “ B Ω i , Ω e : “ R n z Ω i , and ν denotes the unitnormal to Ω i . Recall the definition of the elliptic partial differential expression P from (3.2), the Sobolev space H P p Ω i { e q from (3.6), and the weak conormal derivative B ν from (3.4) and (3.7). For a real valued function α P L p Σ q we define in L p R n q the partial differential operator A α by A α f : “ P f i ‘ P f e , dom A α : “ (cid:32) f “ f i ‘ f e P H P p Ω i q ‘ H P p Ω e q : γf i “ γf e , B ν f e ´ B ν f i “ αγf ( . (4.1)With the help of (3.8) it is not difficult to show that A α is symmetric in L p R n q : Lemma 4.1.
Let α P L p Σ q be real valued. Then the operator A α defined by (4.1) is symmetric in L p R n q .Proof. We show that p A α f, f q L p R n q P R for all f P dom A α . Let f P dom A α befixed. Using (3.8) in Ω i and Ω e and that the normal ν is pointing outside of Ω i andinside of Ω e we get p A α f, f q L p R n q “ p P f i , f i q L p Ω i q ` p P f e , f e q L p Ω e q “ Φ Ω i r f i , f i s ´ p B ν f i , γf i q ` Φ Ω e r f e , f e s ` p B ν f e , γf e q . Since f P dom A α we have γf i “ γf e . This implies, in particular, f P H p R n q and hence Φ Ω i r f i , f i s ` Φ Ω e r f i , f e s “ Φ R n r f, f s . With the help of the transmissioncondition for f P dom A α along Σ we conclude p A α f, f q L p R n q “ Φ R n r f, f s ` p B ν f e ´ B ν f i , γf q “ Φ R n r f, f s ` p αγf, γf q . Since the sesquilinear form Φ R n is symmetric and α is real valued, the latter numberis real and therefore, the claim is shown. (cid:3) In the following proposition we show how the discrete eigenvalues of A α canbe characterized with the help of boundary integral operators. First, we deter-mine the eigenfunctions in ker p A α ´ λ q a ker p A ´ λ q with the Birman-Schwingerprinciple for A α , where the linear eigenvalue problem for the unbounded partialdifferential operator A α is translated to the nonlinear eigenvalue problem for afamily of boundary integral operators which are related to the single layer bound-ary integral operator S p λ q . The eigenfunctions of A α in ker p A α ´ λ q X ker p A ´ λ q are characterized with the help of Theorem 3.8. To formulate the result belowrecall for λ P ρ p A q Y σ disc p A q the definition of the single layer potential SL p λ q from (3.17), the set M λ from (3.18), the single layer boundary integral operator S p λ q from (3.23), S p λ q : “ S p λ q æ L p Σ q , and R A p λ q from (3.35). The followingresult allows us later in Section 4.2 to apply boundary element methods to computeall discrete eigenvalues of A α numerically. Proposition 4.2.
Let α P L p Σ q be real valued and let A α be defined by (4.1) .Then the following is true for any λ P ρ p A q Y σ disc p A q : (i) ker p A α ´ λ q a ker p A ´ λ q ‰ t u if and only if there exists ‰ ϕ P M λ X L p Σ q such that p I ` α S p λ qq ϕ “ . Moreover, (4.2) ker p A α ´ λ qa ker p A ´ λ q “ (cid:32) SL p λ q ϕ : ϕ P M λ X L p Σ q , p I ` α S p λ qq ϕ “ ( . (ii) If λ P ρ p A q , then λ P σ p p A α q if and only if ´ P σ p p α S p λ qq . (iii) ker p A α ´ λ q X ker p A ´ λ q ‰ t u if and only if there exists p ϕ, ψ q J P ran R A p λ q such that αϕ “ . (iv) If λ R σ p p A α q Y σ p A q , then I ` α S p λ q admits a bounded and everywheredefined inverse in L p Σ q .Proof. (i) Assume first that ker p A α ´ λ q a ker p A ´ λ q ‰ t u and let f P ker p A α ´ λ qa ker p A ´ λ q . Then by Lemma 3.3 (i) there exists ϕ P M λ such that f “ SL p λ q ϕ .Since f P dom A α one has with Lemma 3.3 (ii) αγf “ B ν f e ´ B ν f i “ B ν p SL p λ q ϕ q e ´ B ν p SL p λ q ϕ q i “ ´ ϕ. In particular, we deduce ϕ P L p Σ q and with γf “ S p λ q ϕ “ S p λ q ϕ this can berewritten as ´ ϕ “ α S p λ q ϕ . Moreover, the above considerations show(4.3) ker p A α ´ λ q a ker p A ´ λ q Ă (cid:32) SL p λ q ϕ : ϕ P M λ , p I ` α S p λ qq ϕ “ ( . Conversely, assume that there exists 0 ‰ ϕ P M λ X L p Σ q such that p I ` α S p λ qq ϕ “
0. Then f : “ SL p λ q ϕ P H P p R n z Σ q X H p R n q and it follows fromLemma 3.3 (ii) that f is nontrivial. Using the jump properties of SL p λ q ϕ fromLemma 3.3 (ii) we conclude further B ν f e ´ B ν f i “ ´ ϕ “ α S p λ q ϕ “ αγf, where it was used that ϕ belongs to the kernel of I ` α S p λ q . Hence, f P dom A α .With Lemma 3.3 (i) we conclude, as ϕ P M λ , that p A α ´ λ q f “ p P ´ λ qp SL p λ q ϕ q i ‘ p P ´ λ qp SL p λ q ϕ q e “ , which shows λ P σ p p A α q and(4.4) (cid:32) SL p λ q ϕ : ϕ P M λ , p I ` α S p λ qq ϕ “ ( Ă ker p A α ´ λ q . Note that (4.3) and (4.4) give (4.2). Hence, all claims in item (i) are proved.Assertion (ii) is a simple consequence of item (i), as for λ R σ p A q we haveker p A ´ λ q “ t u and M λ “ H ´ { p Σ q .Statement (iii) follows from Theorem 3.8. Note that f P dom A α X dom A ifand only if f P H p R n q and αγf “ B ν f e ´ B ν f i “
0. With Theorem 3.8 it followsthat f P ker p A α ´ λ q X ker p A ´ λ q if and only if there exists p ϕ, ψ q J “ p γf, B ν f q J P ran R A p λ q such that αϕ “ S p λ q is compact in L p Σ q by Lemma 3.4 (i), it follows from Fredholm’salternative that I ` α S p λ q is bijective in L p Σ q and admits a bounded inverse, if0 R σ p p I ` α S p λ qq . According to item (ii) this is fulfilled, if λ R σ p p A α qY σ p A q . (cid:3) Now we are prepared to show the self-adjointness of the operator A α . In theproof of this result we show also a Krein type resolvent formula, which allows usto verify that the essential spectrum of A α coincides with the essential spectrumof the unperturbed operator A . We remark that the resolvent formula in (4.5) iswell defined, as I ` α S p λ q is boundedly invertible in L p Σ q for λ P ρ p A q X ρ p A α q by Proposition 4.2 (iv). Proposition 4.3.
Let α P L p Σ q be real valued, let the operators A , SL p λ q ,and S p λ q , λ P ρ p A q , be given by (3.9) , (3.17) , and (3.23) , respectively, and let S p λ q “ S p λ q æ L p Σ q . Then the operator A α defined by (4.1) is self-adjoint in L p R n q and the following is true: (i) For λ P ρ p A q X ρ p A α q the resolvent of A α is given by (4.5) p A α ´ λ q ´ “ p A ´ λ q ´ ´ SL p λ q ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ . (ii) σ ess p A α q “ σ ess p A q .Proof. In order to prove that A α is self-adjoint, we show that ran p A α ´ λ q “ L p R n q for λ P C zp σ p A q Y σ p p A α qq . Let f P L p R n q be fixed and define g : “ p A ´ λ q ´ f ´ SL p λ q ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ f. Note that g is well defined, as I ` α S p λ q admits a bounded inverse in L p Σ q for λ R σ p A q Y σ p p A α q by Proposition 4.2 (iv). We are going to show that g P dom A α and p A α ´ λ q g “ f . This shows then ran p A α ´ λ q “ L p R n q and also (4.5).Since p A ´ λ q ´ f P H p R n q by Proposition 3.2, we conclude γ p A ´ λ q ´ f P L p Σ q and further from Proposition 4.2 (ii) and Lemma 3.3 thatSL p λ q ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ f P H P p R n z Σ q X H p R n q . Therefore, also g P H P p R n z Σ q X H p R n q . Moreover, we have by Lemma 3.3 (ii) B ν g e ´ B ν g i ´ αγg “ ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ f ´ αγ p A ´ λ q ´ f ` α S p λ q ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ f “ , which shows g P dom A α . Next, we have with ϕ : “ p I ` α S p λ qq ´ αγ p A ´ λ q ´ f p A α ´ λ q g “ p P ´ λ qp A ´ λ q ´ f ´ p P ´ λ qp SL p λ q ϕ q i ‘ p P ´ λ qp SL p λ q ϕ q e “ f, where (3.19) for λ P ρ p A q was used in the last step. With the previous considera-tions we deduce now the self-adjointness of A α and (4.5).It remains to show assertion (ii). Let λ P C z R be fixed. First, due to the mappingproperties of the resolvent of A from Proposition 3.2 and the mapping propertiesof γ from (3.1) the operator γ p A ´ λ q ´ : L p R n q Ñ H { p Σ q (cid:44) Ñ H { p Σ q is bounded. Since H { p Σ q is compactly embedded in L p Σ q this and Proposi-tion 4.2 (iv) yield that ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ : L p R n q Ñ L p Σ q is compact. As L p Σ q is boundedly embedded in H ´ { p Σ q and SL p λ q : H ´ { p Σ q Ñ L p R n q is bounded, we conclude that p A α ´ λ q ´ ´ p A ´ λ q ´ “ ´ SL p λ q ` I ` α S p λ q ˘ ´ αγ p A ´ λ q ´ is compact in L p R n q . Therefore, with the Weyl theorem we get σ ess p A α q “ σ ess p A q . (cid:3) By combining the results from Proposition 4.2 and Proposition 4.3 we can provenow the following proposition about the inverse of the Birman-Schwinger operator I ` α S p λ q , which will be of great importance for the numerical calculation of thediscrete eigenvalues of A α via boundary element methods. Proposition 4.4.
Let α P L p Σ q be real valued and let A α be defined by (4.1) .Then the map ρ p A α q X ρ p A q Q λ ÞÑ ` I ` α S p λ q ˘ ´ can be extended to a holomorphic operator-valued function, which is holomorphicin ρ p A α q with respect to the toplogy in B p L p Σ qq . Moreover, for λ R σ ess p A α q “ σ ess p A q one has ker p A α ´ λ q a ker p A ´ λ q ‰ t u if and only if p I ` α S p λ qq ´ has a pole at λ and (4.6) ker p A α ´ λ qa ker p A ´ λ q “ (cid:32) SL p λ q ϕ : lim λ Ñ λ p λ ´ λ qp I ` α S p λ qq ´ ϕ ‰ ( . Proof.
The proof is split into 4 steps.
Step 1:
Define the map r B ν s Σ : H P p R n z Σ q Ñ H ´ { p Σ q , r B ν s Σ f : “ B ν f i ´ B ν f e , and let λ P ρ p A α q X ρ p A q be fixed. We show that(4.7) ` I ` α S p λ q ˘ ´ “ r B ν s Σ p A α ´ λ q ´ γ ˚ . Note that p I ` α S p λ qq ´ is well defined by the same reasons as in Proposition 4.2 (iv),as α S p λ q P B p H ´ { p Σ q , L p Σ qq is compact in H ´ { p Σ q . In particular, this impliesthat r B ν s Σ p A α ´ λ q ´ γ ˚ P B p H ´ { p Σ qq . To show (4.7) we note first that (4.5)implies γ p A α ´ λ q ´ “ γ p A ´ λ q ´ ´ S p λ q ` I ` α S p λ q ˘ ´ γ p A ´ λ q ´ , which implies, after taking the dual, p A α ´ λ q ´ γ ˚ “ SL p λ q ´ SL p λ q α ` I ` S p λ q α ˘ ´ S p λ q . Using α ` I ` S p λ q α ˘ ´ ´ ` I ` α S p λ q ˘ ´ α “ ` I ` α S p λ q ˘ ´ “` I ` α S p λ q ˘ α ´ α ` I ` S p λ q α ˘‰` I ` S p λ q α ˘ ´ “ , we can simplify the last expression to p A α ´ λ q ´ γ ˚ “ SL p λ q ´ SL p λ q ` I ` α S p λ q ˘ ´ α S p λ q“ SL p λ q ` I ` α S p λ q ˘ ´ “` I ` α S p λ q ˘ ´ α S p λ q ‰ “ SL p λ q ` I ` α S p λ q ˘ ´ . In particular, by Lemma 3.3 the right hand side belongs to B p H ´ { p Σ q , H P p R n z Σ qq and thus, the same must be true for p A α ´ λ q ´ γ ˚ . Therefore, we are allowed toapply r B ν s Σ and the last formula shows, with the help of Lemma 3.3 (ii), therelation (4.7). Step 2:
We show that r B ν s Σ p A α ´ λ q ´ γ ˚ P B p H ´ { p Σ qq for any λ P ρ p A α q andthat ρ p A α q Q λ ÞÑ r B ν s Σ p A α ´ λ q ´ γ ˚ is holomorphic in B p H ´ { p Σ qq .First, we note that dom A α Ă H p R n q X H P p R n z Σ q implies that p A α ´ λ q ´ P B p L p R n q , H p R n qq and p A α ´ λ q ´ P B p L p R n q , H P p R n z Σ qq , see (3.16) for a similar argument. Hence, by duality also p A α ´ λ q ´ P B p H ´ p R n q , L p R n qq . With the resolvent identity this implies for any λ P ρ p A α q and λ P ρ p A α q X ρ p A q ,in a similar way as in the proof of Proposition 3.2, first that r B ν s Σ p A α ´ λ q ´ γ ˚ ´ r B ν s Σ p A α ´ λ q ´ γ ˚ “ p λ ´ λ qr B ν s Σ p A α ´ λ q ´ p A α ´ λ q ´ γ ˚ , which yields first with (4.7) that r B ν s Σ p A α ´ λ q ´ γ ˚ P B p H ´ { p Σ qq and in asecond step, that r B ν s Σ p A α ´ λ q ´ γ ˚ is holomorphic in B p H ´ { p Σ qq , which showsthe claim of this step. Step 3:
With the help of (4.7) and the result from
Step 2 we know that p I ` α S p λ qq ´ can be extended to a holomorphic map in B p H ´ { p Σ qq for λ P ρ p A α q . Byduality we deduce that p I ` α S p λ qq ´ is holomorphic in B p H { p Σ qq for λ P ρ p A α q .Finally, by interpolation we conclude that p I ` α S p λ qq ´ is also holomorphic in B p L p Σ qq for λ P ρ p A α q . Step 4:
Finally, it follows from Proposition 4.2 (i) that ker p A α ´ λ q a ker p A ´ λ q ‰ t u if and only if there exists ϕ P M λ such that p I ` α S p λ qq ϕ “
0, i.e. ifand only if λ ÞÑ p I ` α S p λ qq ´ has a pole at λ . This shows immediately (4.6). (cid:3) Numerical approximation of discrete eigenvalues of A α . For the nu-merical approximation of the discrete eigenvalues of A α and the correspondingeigenfunctions we consider boundary element methods. These require the knowl-edge of an explicit integral representation of the paramatrix G p λ q of P ´ λ orat least a good approximation of the boundary integral operator S p λ q . This isfor example the case when P has constant coefficients. We restrict ourselves tothree-dimensional domains Ω i Ă R in order to keep the presentation simple. Thepresented procedure and the obtained convergence results can be straightforwardlytransfered to domains with general space dimensions. The discrete eigenvalues of A α split into the eigenvalues of the nonlinear eigenvalue problem(4.8) p I ` α S p λ qq ϕ “ ρ p A q and into distinct discrete eigenvalues of A , which can be characterizedon the one hand as poles of the operator-valued functions A p¨q having the propertyspecified in Proposition 4.2 (iii) and on the other hand as poles of r I ` α S p¨qs ´ lying in σ disc p A q .In the following we will first consider the case that there are no discrete eigen-values of A , that means that all discrete eigenvalues can be characterized as eigen-values of the nonlinear eigenvalue problem (4.8). This is for example the casewhen P has constant or periodic coefficients. Afterwards the general case will betreated. For both cases we will present convergence results of the boundary elementapproximations of the discrete eigenvalues of A α . In the first situation a completenumerical analysis is provided, whereas in the general case for the approximation ofthe eigenvalues in σ disc p A q the convergence theory of Section 2 can not be applied.In addition we will address the numerical solution of the discretized problems whichresults in the determination of the poles of matrix-valued functions. For that theso-called contour integral method is suggested [10] which is a reliable method forfinding all poles of a meromorphic matrix-valued function inside a given contour inthe complex plane. Approximation of discrete eigenvalues of A α for the case σ disc p A q “ ∅ . If σ disc p A q “ ∅ , then, by Proposition 4.2 (ii), λ P C z σ ess p A α q is a discrete eigenvalueof A α if and only if it is an eigenvalue of the nonlinear eigenvalue problem (4.8). Anyconforming Galerkin method for the approximation of the eigenvalue problem (4.8)is according to the abstract results in Section 2 a convergent method since ρ p A q Q λ ÞÑ p I ` α S p λ qq is by Lemma 3.4 (iii) holomorphic in B p L p Σ qq and p I ` α S p λ qq satisfies for λ P ρ p A q G˚arding’s inequality of the form (2.1) because α S p λ q : L p Σ q Ñ L p Σ q is compact, see Lemma 3.4 (i).For the presentation of the boundary element method for the approximation ofthe discrete eigenvalues of A α we want to consider first the case that Ω i Ă R is a bounded polyhedral Lipschitz domain. The general case is commented inRemark 4.6. Let p T N q N P N be a sequence of quasi-uniform triangulations of theboundary Σ of Ω i , see e. g. [31, Chapter 4] or [33, Chapter 10], such that(4.9) T N “ t τ N , . . . , τ Nn p N q u and Σ “ n p N q ď j “ τ Nj , where we assume that for the mesh-sizes h p N q of the triangulations T N the relation h p N q Ñ N Ñ 8 . We choose the spaces of piecewise constant functions S p T N q with respect to the triangulations T N as spaces for the approximations ofeigenfunctions of the eigenvalue problem (4.8). For a finite-dimensional subspace V Ă H s p Σ q , s P r , s , we have the following approximation property of S p T N q with respect to } ¨ } L p Σ q [33, Thm. 10.1]:(4.10) δ L p Σ q p V, S p T N qq “ sup v P V } v } L p Σ q “ inf ϕ N P S p T N q } v ´ ϕ N } L p Σ q “ O p h p N q s q . The Galerkin approximation of the eigenvalue problem (4.8) reads as: find eigen-pairs p λ N , ϕ N q P C ˆ S p T N qzt u such that(4.11) pp I ` α S p λ N qq ϕ N , ψ N q “ @ ψ N P S p T N q . All abstract convergence results from Theorem 2.1 can be applied to the approxi-mation of the eigenvalue problem (4.8) by the Galerkin eigenvalue problem (4.11).In the following theorem we only state the asymptotic convergence order of theapproximations of the eigenvalues and the corresponding eigenfunctions.
Theorem 4.5.
Let D Ă ρ p A q be a compact and connected set in C such that B D is a simple rectifiable curve. Suppose that λ P ˚ D is the only eigenvalue of I ` α S p¨q in D and that ker p I ` α S p λ qq Ă H s p Σ q for some s P p , s . Then there exist an N P N and a constant c ą such that for all N ě N we have: (i) For all eigenvalues λ N of the Galerkin eigenvalue problem (4.11) in D (4.12) | λ ´ λ N | ď c p h p N qq ` s holds. (ii) If p λ N , u N q is an eigenpair of (4.11) with λ N P D and } ϕ N } L p Σ q “ , then inf ϕ P ker p I ` α S p λ qq } ϕ ´ ϕ N } L p Σ q ď c p| λ N ´ λ | ` p h p N qq s q . Proof.
The error estimates follow from the abstract convergence results in The-orem 2.1, the approximation property (4.10) of S p T N q , and the fact, that the eigenfunctions of the adjoint problem are more regular than those of p I ` α S p¨qq .To see the last claim, we note that a solution of the adjoint eigenproblem p I ` α S p λ qq ˚ ϕ “ p I ` S p λ q α q ϕ “ H p Σ q and hence, by (4.10) δ L p Σ q ` ker pp I ` α S p λ qq ˚ , S p T N q ˘ ď ch p N q holds. (cid:3) Remark . If Ω is a bounded Lipschitz domain with a curved piecewise C -boundary the approximation of the boundary by a triangulation with flat trianglesas described in [31, Chapter 8] still guarantees convergence of the approximationsof the eigenvalues and eigenfunctions with the same asymptotic convergence orderas in Theorem 4.5. This can be shown by using the results of the discretization ofboundary integral operators for approximated boundaries [31, Chapter 8] and theabstract results of eigenvalue problem approximations [21, 22].The Galerkin eigenvalue problem (4.11) results in a nonlinear matrix eigen-value problem of size n p N q ˆ n p N q , which can be solved by the contour integralmethod [10]. The contour integral method is a reliable method for the approxi-mation of all eigenvalues of a holomorphic matrix-valued function M p¨q which lieinside of a given contour in the complex plane, and for the approximation of thecorresponding eigenvectors. The method is based on the contour integration of theinverse function M p¨q ´ and utilizes that the eigenvalues of the eigenvalue prob-lem for M p¨q are poles of M p¨q ´ . By contour integration of the inverse M p¨q ´ areduction of the holomorphic eigenvalue problem for M p¨q to an equivalent lineareigenvalue problem is possible such that the eigenvalues of the linear eigenvalueproblem coincide with the eigenvalues of the nonlinear eigenvalue problem insidethe contour. For details of the implementation of the method we refer to [10].4.2.2. Approximation of discrete eigenvalues of A α for the case σ disc p A q ‰ ∅ . If σ disc p A q ‰ ∅ , then Proposition 4.2 and Proposition 4.4 show that the discreteeigenvalues of A α are poles of r I ` α S p¨qs ´ or the poles of A p¨q satisfying theproperty specified in Proposition 4.2 (iii). The boundary element approximation ofthe discrete eigenvalues of A α are based on these characterizations.First we want to consider the approximation of the poles of p I ` α S p¨qq ´ . Forthose poles of p I ` α S p¨qq ´ which lie in ρ p A q the abstract convergence results ofSection 2 can be applied with the same reasoning as in the case σ disc p A q “ ∅ , since p I ` α S p¨qq is holomorphic in ρ p A q and the poles of p I ` α S p¨qq ´ in ρ p A q coincidewith the eigenvalues of the eigenvalue problem for p I ` α S p¨qq in ρ p A q . If λ is apole of p I ` α S p¨qq ´ which lies in σ disc p A q , then p I ` α S p¨qq is not holomorphicin λ and therefore the convergence results of Section 2 are not applicable for theboundary element approximation of λ . To the best of our knowledge a rigorousnumerical analysis of the Galerkin approximation of such kind of poles of Fredholmoperator-valued functions for which the inverse is not holomorphic at the poleshave not been considered so far in the literature. However, we expect similarconvergence results also of such kind of poles. If this holds, then this kind ofpoles of p I ` α S p λ qq ´ , which is holomorphic in ρ p A α q by Proposition 4.4, areappropriately represented as poles of the discretized problem and will be identifiedby the contour integral method. Finally, we want to discuss the approximation of the discrete eigenvalues of A α which are not poles of r I ` α S p¨qs ´ . If λ is such an eigenvalue, then, byProposition 4.2 (iii), it is a pole of A p¨q such that a pair p ϕ, ψ q P ran R A p λ q definedby (3.35) exists with αϕ “ p λ, ϕ, ψ q , p ϕ, ψ q ‰ p , q , satisfies(4.13) A p λ q ´ ˆ ψϕ ˙ “ ˆ ˙ and αϕ “ . The characterization in (4.13) can be used for the numerical approximation of thediscrete eigenvalues of A α which are not poles of r I ` α S p¨qs ´ . For the boundaryelement approximation of the eigenvalue problem in (4.13) we need in additionto the space of piecewise constant functions S p T N q the space of piecewise linearfunctions S p T N q . Formally, the Galerkin eigenvalue problem(4.14) ˆ A p λ q ´ ˆ ψ N ϕ N ˙ , ˆ r ψ N r ϕ N ˙˙ “ ˆ r ψ N r ϕ N ˙ P S p T N q ˆ S p T N q is considered. However, if the contour integral method is used for the computationsof the eigenvalues of the Galerkin eigenvalue problem (4.14), then A p¨q ´ has notto be computed, since the contour integral method operates on its inverse A p¨q .The abstract convergence results of Section 2 can be applied to the approximationof those eigenvalues λ of the eigenvalue problem (4.13) for which A p¨q ´ is holo-morphic. In general it is possible that λ is a pole of A p¨q and of A p¨q ´ . In thiscase, as mentioned before, a rigorous analysis of the Galerkin approximation hasnot been provided so far.4.3. Numerical examples.
We present two numerical examples for P “ ´ ∆. Inthis case A is the free Laplace operator and σ p A q “ σ ess p A q “ r , , and thefundamental solution for P ´ λ is given by G p λ ; x, y q “ e i ? λ } x ´ y } p π } x ´ y }q ´ [28,Chapter 9]. In particular, the operator A has no discrete eigenvalues and thereforethe eigenvalues of A α coincide with the eigenvalues of the eigenvalue problem for I ` α S p¨q . The Galerkin eigenvalue problem (4.11) is used for the computation ofapproximations of discrete eigenvalues of A α and corresponding eigenfunctions. Inall numerical experiments the open-source library BEM++ [32] is employed for thecomputations of the boundary element matrices.4.3.1. Unit ball.
As first numerical example we consider as domain Ω i the unitball and a constant α . The eigenvalues of A α for constant α have an analyticalrepresentation [2, Theorem 3.2] which are used to show that in the numerical ex-periments the predicted convergence order (4.12) is reflected. Let l P N be suchthat 2 l ` ă ´ α . Then λ p l q is an eigenvalue of A α of multiplicity 2 l ` ` αI l ` { `a ´ λ p l q ˘ K l ` { `a ´ λ p l q ˘ “ , where I l ` { and K l ` { denote modified Bessel functions of order l ` {
2. Con-versely, all eigenvalues of A α are of the above form.For the numerical experiments we choose α “ ´
6. In Table 1 the errors ofthe approximation of the eigenvalues of A α with α “ ´ h are given. For multiple eigenvalues λ p l q , l “ ,
2, we have used the meanvalue of the approximations, denoted by p λ p l q h , for the computation of the error. Theexperimental convergence order (eoc) reflects the predicted quadratic convergence order (4.12). In Figure 1 plots of computed eigenfunctions of A α in the xy -plane aregiven where for each exact eigenvalue one approximated eigenfunction is selected. h ˇˇˇ λ p q h ´ λ p q ˇˇˇ | λ p q | eoc ˇˇˇp λ p q h ´ λ p q ˇˇˇ | λ p q | eoc ˇˇˇp λ p q h ´ λ p q ˇˇˇ | λ p q | eoc0.2 1.203e-2 - 2.837e-2 - 1.666e-1 -0.1 2.473e-3 2.28 6.968e-3 2.02 3.969e-2 2.070.05 4.344e-4 2.48 1.781e-3 1.95 9.593e-3 2.06 Table 1.
Error of the approximations of the eigenvalues of A α , α “ ´
6, of the unit sphere for different mesh-sizes h . Figure 1.
Computed eigenfunctions of A α , α “ ´
6, in the xy -plane for the unit ball.4.3.2. Screen.
For the second numerical example we have chosen a δ -potential sup-ported on the non-closed surface Γ : “ r , s ˆ r , s ˆ t u Ă R , which is referredto as screen. The interaction strength α is defined by α “ ´ χ Γ , where χ Γ is thecharacteristic function on Γ given as χ Γ p x q : “ , for x P Γ , , else . Such a problem fits in the described theory of this section. Take for example asdomain Ω i the unit cube, as we have done in our numerical experiments, then Γ isidentical with one of the faces of Σ “ B Ω i .In the numerical experiments we have chosen as contour the ellipse g p t q “ c ` a cos p t q ` ib sin p t q , t P r , π s , with c “ ´ . a “ .
99 and b “ .
01. We have gotfour eigenvalues of the discretized eigenvalue problem inside the contour, namely λ p q h “ ´ . λ p q h “ ´ . λ p q h “ ´ .
88, and λ p q h “ ´ .
59 for the mesh-size h “ . xy -plane are given in Figure 2.5. Elliptic differential operators with δ -interactions supported oncompact Lipschitz smooth surfaces In this section we study the spectral properties of the partial differential operatorwhich corresponds to the formal expression B β : “ P ` β x δ Σ , ¨y δ in a mathemat-ically rigorous way and study its spectral properties. The considerations are very Figure 2.
Computed eigenfunctions of A α in the xy -plane for α “ ´ χ r , sˆr , sˆt u .similar as for A α in Section 4. First, in Section 5.1 we show the self-adjointness of B β in L p R n q and obtain the Birman-Schwinger principle to characterize the dis-crete eigenvalues of B β via boundary integral operators in Proposition 5.2. Then,in Section 5.2 we discuss how these boundary integral equations can be solved nu-merically by boundary element methods. Finally, in Section 5.3 we show somenumerical examples.5.1. Definition and self-adjointness of B β . For a real valued function β with β ´ P L p Σ q we define in L p R n q the partial differential operator B β by B β f : “ P f i ‘ P f e , dom B β : “ (cid:32) f “ f i ‘ f e P H P p Ω i q ‘ H P p Ω e q : B ν f i “ B ν f e , γf e ´ γf i “ β B ν f ( . (5.1)With the help of (3.8) it is not difficult to show that B β is symmetric in L p R n q : Lemma 5.1.
Let β be a real valued function on Σ with β ´ P L p Σ q . Then theoperator B β defined by (5.1) is symmetric in L p R n q .Proof. We show that p B β f, f q L p R n q P R for all f P dom B β . Let f P dom B β befixed. Using (3.8) in Ω i and Ω e and that the normal ν is pointing outside of Ω i andinside of Ω e we get p B β f, f q L p R n q “ p P f i , f i q L p Ω i q ` p P f e , f e q L p Ω e q “ Φ Ω i r f i , f i s ´ p B ν f i , γf i q ` Φ Ω e r f e , f e s ` p B ν f e , γf e q . Since f P dom B β we have B ν f i “ B ν f e and β B ν f “ p γf e ´ γf i q . Therefore, weconclude p B β f, f q L p R n q “ Φ Ω i r f i , f i s ` Φ Ω e r f e , f e s ` p B ν f, γf e ´ γf i q“ Φ Ω i r f i , f i s ` Φ Ω e r f e , f e s ` p B ν f, β B ν f q . Since the sesquilinear forms Φ Ω i { e are symmetric, the latter number is real andtherefore, the claim is shown. (cid:3) The following proposition is the counterpart of Proposition 4.2 to characterizethe discrete eigenvalues of B β via boundary integral operators. It is the theoreticbasis to compute these eigenvalues with the help of boundary element methods inSection 5.2. To formulate the result below recall for λ P ρ p A q Y σ disc p A q the defi-nition of the double layer potential DL p λ q from (3.25), the set N λ from (3.26), thehypersingular boundary integral operator R p λ q from (3.31), and R A p λ q from (3.35). Proposition 5.2.
Let β be a real valued function on Σ with β ´ P L p Σ q and let B β be defined by (5.1) . Then the following is true for any λ P ρ p A q Y σ disc p A q : (i) ker p B β ´ λ q a ker p A ´ λ q ‰ t u if and only if there exists ‰ ϕ P N λ suchthat p β ´ ` R p λ qq ϕ “ . Moreover, (5.2) ker p B β ´ λ q a ker p A ´ λ q “ (cid:32) DL p λ q ϕ : ϕ P N λ , p β ´ ` R p λ qq ϕ “ ( . (ii) If λ P ρ p A q , then λ P σ p p B β q if and only if P σ p p β ´ ` R p λ qq . (iii) ker p B β ´ λ q X ker p A ´ λ q ‰ t u if and only if there exists p ϕ, ψ q J P ran R A p λ q such that ψ “ . (iv) If λ R σ p p B β q Y σ p A q , then β ´ ` R p λ q : H { p Σ q Ñ H ´ { p Σ q admits abounded and everywhere defined inverse.Proof. (i) Assume first that ker p B β ´ λ q a ker p A ´ λ q ‰ t u and let f P ker p B β ´ λ qa ker p A ´ λ q . Then by Lemma 3.5 (i) there exists ϕ P N λ such that f “ DL p λ q ϕ .Since f P dom B β one has with Lemma 3.5 (ii) β B ν f “ γf e ´ γf i “ γ p DL p λ q ϕ q e ´ γ p DL p λ q ϕ q i “ ϕ. With B ν f “ ´ R p λ q ϕ this can be rewritten as β ´ ϕ “ ´ R p λ q ϕ . Hence, the aboveconsiderations show(5.3) ker p B β ´ λ q a ker p A ´ λ q Ă (cid:32) DL p λ q ϕ : ϕ P N λ , p β ´ ` R p λ qq ϕ “ ( . Conversely, assume that there exists ϕ P N λ such that p β ´ ` R p λ qq ϕ “ f : “ DL p λ q ϕ P H P p R n z Σ q is nontrivial by Lemma 3.5 (ii). Using the jumpproperties of DL p λ q ϕ from Lemma 3.5 (ii) we conclude further B ν f i “ B ν f e and γf e ´ γf i “ γ p DL p λ q ϕ q e ´ γ p DL p λ q ϕ q i “ ϕ “ ´ β R p λ q ϕ “ β B ν f, where β ´ p I ` β R p λ qq ϕ “ f P dom B β . With Lemma 3.5 (i)we conclude with ϕ P N λ eventually p B β ´ λ q f “ p P ´ λ qp DL p λ q ϕ q i ‘ p P ´ λ qp DL p λ q ϕ q e “ , which shows λ P σ p p B β q and(5.4) (cid:32) DL p λ q ϕ : ϕ P N λ , p β ´ ` R p λ qq ϕ “ ( Ă ker p B β ´ λ q a ker p A ´ λ q . Note that (5.3) and (5.4) give (5.2). Hence, all claims in item (i) are proved.Assertion (ii) is a simple consequence of item (i), as for λ R σ p A q we haveker p A ´ λ q “ t u and N λ “ H { p Σ q .Statement (iii) follows from Theorem 3.8. Note that f P dom B β X dom A if andonly if f P H p R n q and B ν f “ β ´ p γf e ´ γf i q “
0. With Theorem 3.8 it followsthat f P ker p B β ´ λ q X ker p A ´ λ q if and only if there exists p ϕ, ψ q J “ p γf, B ν f q J P ran R A p λ q such that ψ “ β ´ P L p Σ q givesrise to a bounded operator from H { p Σ q to L p Σ q and as L p Σ q is compactly em-bedded in H ´ { p Σ q , the operator β ´ : H { p Σ q Ñ H ´ { p Σ q is compact. There-fore, we deduce from [28, Theorem 2.26] that β ´ ` R p λ q : H { p Σ q Ñ H ´ { p Σ q is a Fredholm operator with index zero, as R p λ q is a Fredholm operator with in-dex zero by Lemma 3.6 (ii). Since λ is not an eigenvalue of B β by assumption,we deduce from (ii) that β ´ ` R p λ q is injective and hence, this operator is alsosurjective. Therefore, it follows from the open mapping theorem that β ´ ` R p λ q has a bounded inverse from H ´ { p Σ q to H { p Σ q . (cid:3) In the following proposition we show the self-adjointness of B β and a Krein typeresolvent formula for this operator. We remark that the resolvent formula in (5.5)is well defined, as β ´ ` R p λ q : H { p Σ q Ñ H ´ { p Σ q is boundedly invertible for λ P ρ p A q X ρ p B β q by Proposition 5.2 (iv). Proposition 5.3.
Let β be a real valued function on Σ with β ´ P L p Σ q and letthe operators A , DL p λ q , and R p λ q , λ P ρ p A q , be given by (3.9) , (3.25) , and (3.31) ,respectively. Then the operator B β defined by (5.1) is self-adjoint in L p R n q andthe following is true: (i) For λ P ρ p A q X ρ p B β q the resolvent of B β is given by (5.5) p B β ´ λ q ´ “ p A ´ λ q ´ ` DL p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ . (ii) σ ess p B β q “ σ ess p A q .Proof. In order to show that B β is self-adjoint, we show that ran p B β ´ λ q “ L p R n q for λ P C zp σ p A q Y σ p p B β qq . Let f P L p R n q be fixed and define g : “ p A ´ λ q ´ f ` DL p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ f. Note that g is well defined, as β ´ ` R p λ q : H { p Σ q Ñ H ´ { p Σ q admits a boundedinverse for λ R σ p A q Y σ p p B β q by Proposition 5.2 (iv). We are going to show that g P dom B β and p B β ´ λ q g “ f . This shows then ran p B β ´ λ q “ L p R n q andalso (5.5).Since p A ´ λ q ´ f P H p R n q by Proposition 3.2 implies B ν p A ´ λ q ´ f P L p Σ q Ă H ´ { p Σ q , we conclude from Proposition 5.2 (iv) and Lemma 3.5 thatDL p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ f P H P p R n z Σ q . Therefore, also g P H P p R n z Σ q . Moreover, we get with the help of Lemma 3.5 (ii)that B ν g e “ B ν g i . Applying once more Lemma 3.5 (ii) we conclude β ´ p γg e ´ γg i q ´ B ν g “ β ´ ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ f ´ B ν p A ´ λ q ´ f ` R p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ f “ , which shows g P dom B β . Next, we have with ϕ : “ p β ´ ` R p λ qq ´ B ν p A ´ λ q ´ f p B β ´ λ q g “ p P ´ λ qp A ´ λ q ´ f ` p P ´ λ qp DL p λ q ϕ q i ‘ p P ´ λ qp DL p λ q ϕ q e “ f, where (3.27) was used in the last step. With the previous considerations we deducenow the self-adjointness of B β and (5.5).It remains to show assertion (ii). Let λ P C z R be fixed. Due to the mappingproperties of the resolvent of A from Proposition 3.2 and the mapping propertiesof B ν from (3.7) the operator B ν p A ´ λ q ´ : L p R n q Ñ L p Σ q (cid:44) Ñ H ´ { p Σ q is bounded. Hence, Proposition 5.2 (iv) yields that ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ : L p R n q Ñ H { p Σ q is bounded. As H { p Σ q is compactly embedded in L p Σ q we conclude that thelatter operator is compact from L p R n q to L p Σ q . Since DL p λ q : L p Σ q Ñ L p R n q is bounded by (3.25), we find eventually that p B β ´ λ q ´ ´ p A ´ λ q ´ “ DL p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´
12 M. HOLZMANN AND G. UNGER is compact in L p R n q . Therefore, we get with the Weyl theorem σ ess p B β q “ σ ess p A q . (cid:3) The following result is the counterpart of Proposition 4.4 on the inverse of theBirman-Schwinger operator β ´ ` R p λ q , which will be of great importance forthe numerical calculation of the discrete eigenvalues of B β via boundary elementmethods. Proposition 5.4.
Let β be a real valued function with β ´ P L p Σ q and let B β bedefined by (5.1) . Then the map ρ p B β q X ρ p A q Q λ ÞÑ ` β ´ ` R p λ q ˘ ´ can be extended to a holomorphic operator-valued function, which is holomorphicin ρ p B β q with respect to the toplogy in B p H ´ { p Σ q , H { p Σ qq . Moreover, for λ R σ ess p B β q “ σ ess p A q one has ker p B β ´ λ q a ker p A ´ λ q ‰ t u if and only if p β ´ ` R p λ qq ´ has a pole at λ and (5.6) ker p B β ´ λ qa ker p A ´ λ q “ (cid:32) DL p λ q ϕ : lim λ Ñ λ p λ ´ λ qp β ´ ` R p λ qq ´ ϕ ‰ ( . Proof.
The proof is similar as the proof of Proposition 4.4 and split into 3 steps.
Step 1:
Define the map r γ s Σ : H P p R n z Σ q Ñ H { p Σ q , r γ s Σ f : “ γf e ´ γf i . Let λ P ρ p B β q X ρ p A q be fixed. We show that(5.7) ` β ´ ` R p λ q ˘ ´ “ r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ . In particular, with Proposition 5.2 (iv) this implies that r γ s Σ p B β ´ λ q ´ r γ s Σ belongsto B p H ´ { p Σ q , H { p Σ qq . To show (5.7) we note first that r γ s Σ f “ β B ν f holds for f P dom B β and hence (5.5) implies r γ s Σ p B β ´ λ q ´ “ β B ν p B β ´ λ q ´ “ β B ν p A ´ λ q ´ ´ β R p λ q ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ “ β “` β ´ ` R p λ q ˘ ´ R p λ q ‰` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ “ ` β ´ ` R p λ q ˘ ´ B ν p A ´ λ q ´ , which implies, after taking the dual, p B β ´ λ q ´ r γ s ˚ Σ “ DL p λ q ` β ´ ` R p λ q ˘ ´ . In particular, by Lemma 3.5 and Proposition 5.2 (iv) the right hand side belongsto B p H ´ { p Σ q , H P p R n z Σ qq and thus the same must be true for p B β ´ λ q ´ r γ s ˚ Σ .Therefore, we are allowed to apply r γ s Σ and the last formula shows, with the helpof Lemma 3.5 (ii), the relation (5.7). Step 2:
We show that r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ P B p H ´ { p Σ q , H { p Σ qq for any λ P ρ p B β q and that the mapping ρ p B β q Q λ ÞÑ r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ is holomorphicin B p H ´ { p Σ q , H { p Σ qq .First, we note that dom B β Ă H P p R n z Σ q implies that p B β ´ λ q ´ P B p L p R n q , H P p R n z Σ qq , see (3.16) for a similar argument. Hence, by duality also p B β ´ λ q ´ P B pp H P p R n z Σ qq ˚ , L p R n qq . With the resolvent identity this implies for any λ P ρ p B β q and λ P ρ p B β q X ρ p A q ,in a similar way as in the proof of Proposition 3.2, that r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ ´ r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ “ p λ ´ λ qr γ s Σ p B β ´ λ q ´ p B β ´ λ q ´ r γ s ˚ Σ , which shows first with (5.7) that r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ P B p H ´ { p Σ q , H { p Σ qq andin a second step, that r γ s Σ p B β ´ λ q ´ r γ s ˚ Σ is holomorphic in B p H ´ { p Σ q , H { p Σ qq ,which shows the claim of this step. Step 3:
Finally, it follows from Proposition 5.2 (i) that ker p B β ´ λ q a ker p A ´ λ q ‰ t u if and only if there exists ϕ P N λ such that p β ´ ` R p λ qq ϕ “
0, i.e. ifand only if λ ÞÑ p β ´ ` R p λ qq ´ has a pole at λ . This shows immediately (5.6). (cid:3) Numerical approximation of discrete eigenvalues of B β . The approx-imation of the discrete eigenvalues of B β by boundary element methods is basedon the same principles as those for the discrete eigenvalues of A α described in Sec-tion 4.2. In order to apply boundary element methods for the approximation of thediscrete eigenvalues of B β it is necessary to have an integral representation of theparamatrix G p λ q of P ´ λ or at least a good approximation of the boundary inte-gral operator R p λ q . We use the characterization of the discrete eigenvalues of B β interms of boundary integral operators given in Proposition 5.2 and Proposition 5.4.The discrete eigenvalues split into the eigenvalues of the nonlinear eigenvalue prob-lem(5.8) p β ´ ` R p λ qq ψ “ ρ p A q and into distinct discrete eigenvalues of A which are either the poles of A p¨q satisfying the properties specified in Proposition 5.2 (iii) or poles of p β ´ ` R p¨qq ´ in σ disc p A q .In the following presentation of the boundary element method we want to con-sider first the case that σ disc p A q “ ∅ and then the general case. If σ disc p A q “ ∅ ,then the discrete eigenvalues of B β coincide with the eigenvalues of the nonlineareigenvalue problem (5.8) in ρ p A q as shown in Proposition 5.2 (ii). In this situationa complete convergence analysis is provided by the theory of Section 2. For thegeneral case the convergence of the approximations of the discrete eigenvalues of B β which lie in σ disc p A q is an open issue.The discretized problems for the approximation of the discrete eigenvalues of B β which result from the approximations of the boundary integral operators by bound-ary element methods are problems for the determination of poles of meromorphicmatrix-valued functions. For this kind of problems we suggest the contour integralmethod [10], which was discussed in Section 4.2.1.5.2.1. Approximation of discrete eigenvalues of B α for the case σ disc p A q “ ∅ . For σ disc p A q “ ∅ the discrete eigenvalues of B β coincide, according to Propo-sition 5.2 (ii), with the eigenvalues of the eigenvalue problem for p β ´ ` R p¨qq .Lemma 3.6 (iii) shows that the map ρ p A q Q λ ÞÑ p β ´ ` R p λ qq is holomorphicin B p H { p Σ q , H ´ { p Σ qq . Moreover, by Lemma 3.6 (i) the operators β ´ ` R p λ q satisfy for λ P ρ p A q G˚arding’s inequality of the form (2.1). Hence, any conform-ing Galerkin method for the approximation of the eigenvalue problem (5.8) is aconvergent method, which follows from the theory in Section 2.For the boundary element approximation of the eigenvalue problem (5.8) we firstassume that Ω i is a polyhedral Lipschitz domain. The case of general Lipschitz do-mains is addressed in Remark 5.6. Let p T N q N P N be a sequence of quasi-uniformtriangulations of Ω i with the properties specified in (4.9). As approximation spacefor the approximation of the eigenfunctions of the eigenvalue problem (5.8) wechoose the space S p T N q of piecewise linear functions with respect to the triangu-lation T N . The approximation property of S p T N q depends on the regularity of thefunctions which are approximated. In order to measure the regularity of functionsdefined on a piecewise smooth boundary Σ, partitioned by open sets Σ , . . . , Σ J such that Σ “ J ď j “ Σ j , Σ j X Σ i “ ∅ for i ‰ j, so-called piecewise Sobolev spaces of order s ą H s pw p Σ q : “ t v P H p Σ q : v æ Σ j P H s p Σ j q for j “ , . . . , J u are used, see [31, Definition 4.1.48]. For s P r , s the space H s pw p Σ q is defined by H s pw p Σ q : “ H s p Σ q . If W is a finite dimensional subspace of H { ` s pw p Σ q for s P p , s ,then(5.9) δ H { p Σ q p W, S p T N qq “ sup w P W } w } H { p Σ q “ inf ψ N P S p T N q } w ´ ψ N } H { p Σ q “ O p h p N q s q holds [31, Proposition 4.1.50].The Galerkin approximation of the eigenvalue problem (5.8) reads as follows:find eigenvalues λ N P C and corresponding eigenfunctions ψ N P S p T N qzt u suchthat(5.10) ` p β ´ ` R p λ N qq ψ N , χ N ˘ “ @ χ N P S p T N q . We can apply all convergence results from Theorem 2.1 to the Galerkin eigenvalueproblem (5.10). In the following theorem the asymptotic convergence order ofthe approximations of the eigenvalues and the corresponding eigenfunctions arespecified.
Theorem 5.5.
Let D Ă ρ p A q be a compact and connected set in C such that B D isa simple rectifiable curve. Suppose that λ P ˚ D is the only eigenvalue of p β ´ ` R p¨qq in D and that ker p β ´ ` R p λ qq Ă H { ` s pw p Σ q for some s P p , s . Then there existan N P N and a constant c ą such that for all N ě N we have: (i) For all eigenvalues λ N of the Galerkin eigenvalue problem (5.10) in D (5.11) | λ ´ λ N | ď c p h p N qq s holds. (ii) If p λ N , u N q is an eigenpair of (5.10) with λ N P D and } ψ N } H { p Σ q “ ,then (5.12) inf ψ P ker p β ´ ` R p λ qq } ψ ´ ψ N } H { p Σ q ď c p| λ N ´ λ | ` p h p N qq s q . Proof.
We have shown in Lemma 3.6 (iii) that the map ρ p A q Q λ ÞÑ p β ´ ` R p λ qq isholomorphic in B p H { p Σ q , H ´ { p Σ qq . Moreover, by Lemma 3.6 (i), the operators β ´ ` R p λ q satisfy for λ P ρ p A q G˚arding’s inequality of the form (2.1). TheGalerkin approximation (5.10) of the eigenvalue problem for p β ´ ` R p¨qq is aconforming approximation since S p T N q is a subspace of H { p Σ q . Hence, we canuse Theorem 2.1. The error estimates follows from the approximation property (5.9)of S p T N q and the fact, that the eigenfunction of the adjoint problem are as regularas for p β ´ ` R p¨qq . (cid:3) Remark . If Ω is a bounded Lipschitz domain with a curved piecewise C -boundary the approximation of the boundary by a triangulation with flat trianglesas described in [31, Chapter 8] reduces the maximal possible convergence order s for the error of the eigenvalues in (5.11) and for the error of the eigenfunctionsin (5.12) from s “ to s “
1. This follows from the results of the discretization ofboundary integral operators for approximated boundaries [31, Chapter 8] and fromthe abstract results of eigenvalue problem approximations [21, 22].5.2.2.
Approximation of discrete eigenvalues of B α for the case σ disc p A q ‰ ∅ . If σ disc p A q ‰ ∅ , then Proposition 5.2 and Proposition 5.4 imply that the discreteeigenvalues of B β are poles of p β ´ ` R p¨qq ´ or poles of A p¨q with the property givenin Proposition 5.2 (iii). These characterizations are used for the approximation ofthe discrete eigenvalues of B β . We will separately discuss both cases.Let λ be a discrete eigenvalue of B β and in addition be a pole of p β ´ ` R p¨qq ´ .Then p β ´ ` R p¨qq is either holomorphic in λ , which is the case for λ P ρ p A q , or λ is a pole of p β ´ ` R p¨qq . A pole λ P ρ p A q of p β ´ ` R p¨qq ´ can be consideredas an eigenvalue of the eigenvalue problem for the homomorphic Fredholm operator-valued function p β ´ ` R p¨qq in ρ p A q and the convergence results of Section 2 canbe applied with the same reasoning as in the case of σ disc p A q “ ∅ . If λ P σ disc p A q is a pole of p β ´ ` R p¨qq , then the convergence theory of Section 2 is not applicablefor λ . We expect convergence of the approximations for this kind of poles of p β ´ ` R p¨qq ´ , but a rigorous numerical analysis has not established so far.The approximation of a discrete eigenvalue λ of B β which is not a pole of p β ´ ` R p¨qq ´ is based on the following characterization from Proposition 5.2 (iii): λ is a pole of A p¨q and there exists a pair p , q ‰ p ψ, ϕ q P H { p Σ q ˆ H ´ { p Σ q such that(5.13) A p λ q ´ ˆ ψϕ ˙ “ ˆ ˙ and ψ “ . For the approximation of the eigenvalues of the nonlinear eigenvalue problem in (5.13)formally the Galerkin problem in S p T N q ˆ S p T N q as given in (4.14) is considered.If the contour integral method is used for the computations of the approximationsof the eigenvalues for A p¨q ´ , then A p¨q ´ does not have not be computed, butinstead its inverse A p¨q . The convergence theory of Section 2 can be applied tothe approximation of those eigenvalues of A p¨q ´ for which A p¨q ´ is holomorphic.If λ is a pole A p¨q and of A p¨q ´ , we again expect convergence, but a numericalanalysis for such kind of poles has not been provided so far.5.3. Numerical examples.
For the numerical examples of the approximation ofdiscrete eigenvalues of B β we choose P “ ´ ∆. In this case σ ess p A q “ r , , σ disc p A q “ ∅ , the fundamental solution is given by G p λ ; x, y q “ e i ? λ } x ´ y } p π } x ´ y }q ´ , and the discrete eigenvalues of B β coincide with the eigenvalues of the non-linear eigenvalue problem for p β ´ ` R p¨qq . The Galerkin eigenvalue problem (5.10)is used for the computation of approximations of discrete eigenvalues of B β andcorresponding eigenfunctions.5.3.1. Unit ball.
We consider for the first numerical example as domain Ω i again theunit ball. Analytical representations for the discrete eigenvalues of B β are knownin this case [2, Section 6] and are used to compute the errors of the approximationsand to check the predicted asymptotic error estimate (5.11). The errors of theapproximations of the eigenvalues of B β with β ´ “ ´ . g p t q “ c ` a cos p t q ` ib sin p t q , t P r , π s , with c “ ´ . a “ .
99 and b “ .
01 are given in Table 2 for three different mesh sizes h . We denote by p λ p l q h , l “ ,
2, the mean value of the approximations of the multiple eigenvalues λ p l q . Aquadratic experimental convergence order (eoc) can be observed which is accordingto Remark 5.6 the best possible convergence order if flat triangles are used for thetriangulation of a curved boundary as it has been done in our experiments. InFigure 3 plots of computed eigenfunctions of B β in the xy -plane are given wherefor each exact eigenvalue one approximated eigenfunction is selected. h ˇˇˇ λ p q h ´ λ p q ˇˇˇ | λ p q | eoc ˇˇˇp λ p q h ´ λ p q ˇˇˇ | λ p q | eoc ˇˇˇp λ p q h ´ λ p q ˇˇˇ | λ p q | eoc0.2 3.232e-3 - 1.885e-3 - 6.745e-3 -0.1 7.099e-4 2.19 3.926e-4 2.26 1.406e-3 2.260.05 1.635e-4 2.11 8.958e-5 2.13 3.054e-4 2.20 Table 2.
Error of the approximations of the eigenvalues of B β , β ´ “ ´ .
5, of the unit ball for different mesh-sizes h . Figure 3.
Computed eigenfunctions of B β , β ´ “ ´ .
5, in the xy -plane for the unit ball.5.3.2. L-shape domain.
In the second numerical example we have chosen as domainΩ i a so-called L-shape domain with Ω i “ p´ , q zpr , s ˆ r´ , sq and we haveset β ´ “ ´ .
75. In the numerical experiments the ellipse g p t q “ c ` a cos p t q ` ib sin p t q , t P r , π s , with c “ ´ . a “ .
99 and b “ .
01, is taken as contourfor the contour integral method. We have got three eigenvalues of the discretized Figure 4.
Computed eigenfunctions of B β , β ´ “ ´ .
75, for theL-shape domain Ω i “ p´ , q zpr , s ˆ r´ , sq in the xy -plane.eigenvalue problem inside this contour, namely λ p q h “ ´ . λ p q h “ ´ .
41 and λ p q h “ ´ .
94 for the mesh-size h “ .
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