The exterior Calderón operator for non-spherical objects

This paper deals with the exterior Calderón operator for not necessarily spherical domains. We present a new approach of finding the norm of the exterior Calderón operator for a wide class of surfaces. The basic tool in the treatment is the set of eigenfunctions and eigenvalues to the Laplace–Beltrami operator for the surface. The norm is obtained in view of an eigenvalue problem of a quadratic form containing the exterior Calderón operator. The connection of the exterior Calderón operator to the transition matrix for a perfectly conducting surface is analyzed.


Introduction
The exterior Calderón operator maps the tangential scattered electric surface field to the corresponding magnetic surface field. This operator is also called the Poincaré-Steklov operator, and its discretization is often called the Schur complement. It has been studied intensively during many years, see e.g., [9,18,20].
It is related to the Dirichlet-to-Neumann map for the scalar Helmholtz equation. The exterior Calderón map is instrumental in the analysis of the solution to the exterior solution of the scattering problem. In fact, it is strongly related to the solution of the scattering problem by a perfectly conducting (PEC) obstacle, which is a subject we analyze in Sect. 5.
The norm of the exterior Calderón operator determines the largest amplification factor of the surface fields. This norm specifies the largest impedance (the quotient between scattered tangential magnetic and electric fields) that can exist for a given scattering geometry. In several numerical implementations of the scattering problem, such as the Methods of Moments (MoM), the impedance matrix represents the exterior Calderón operator and this matrix is instrumental for the numerical solution of the problem. This observation gives a physical interpretation of the value of the norm of the exterior Calderón operator.
A new way of finding this norm is presented in this paper. The key ingredient in this analysis is the set of eigenfunctions to the Laplace-Beltrami operator of the surface. These eigenfunctions and the corresponding eigenvalues are intrinsic to the surface and constitute an excellent tool for further analysis; the literature on this subject of finding these eigenfunctions and eigenvalues is extensive, see, e.g., [4,11,19,27]. Explicit values of the norm of the exterior Calderón operator have only been obtained for the sphere case [18,20] and the planar case [3,9], and we refer to these bibliographical items for the explicit techniques of computing the norm. In this paper, we present a new way to explicitly find the norm for non-spherical obstacles. The final expression of the norm for a non-spherical obstacle is related to an eigenvalue problem of a quadratic form containing the exterior Calderón matrix.
An outline of the organization of the contents in this paper is now presented. In Sect. 2, the statement of the problem is introduced, the exterior Calderón operator is defined, and the useful integral representation of the scattered field is presented. The intrinsic generalized harmonics (both scalar and vector valued) are introduced in Sect. 3, and these functions are used in Sect. 4. The generalized harmonics developed in Sect. 3 constitute a great asset, and they serve as a natural orthonormal basis for the expansion of the surface fields in many scattering problems. A matrix representation of the exterior Calderón problem in terms of the generalized harmonics is presented in Sect. 4, and this matrix has many valuable properties that are useful in the solution of the exterior scattering problem. Section 4 also contains a constructive method to compute the norm of the exterior Calderón operator for non-spherical obstacles. The connection between the exterior Calderón operator and the transition matrix of the corresponding perfectly conducting obstacle is clarified in Sect. 5. The spherical geometry is explicitly treated in Sect. 6. The paper is concluded with some final remarks in Sect. 7.

Formulation of the scattering problem
In this section, we present the geometry of the problem and the solution of the scattered field in the exterior region.
The Maxwell equations in the exterior region are given by 4 (we adopt the time convention e Àixt ) r Â EðxÞ ¼ ikHðxÞ The wave number k ¼ x=c is assumed to be a positive constant, where x is the angular frequency of the fields, and c is the speed of light in the exterior medium. In the region X e , the (scattered) fields satisfy the time-harmonic Maxwell equations (1) and the Silver-Müller radiation condition at infinity, and we are looking for solutions E s and H s in the space H loc ðcurl; X e Þ.
The trace operators p and c on CðX e Þ are given by pðuÞ ¼m Â ðuj oX ÂmÞ and cðuÞ ¼m Â uj oX , respectively, 5 and in the case that u belongs to H loc ðcurl; X e Þ, the fields have traces on oX belonging to H À1=2 ðdiv; CÞ; more precisely we have cðE s Þ; cðH s Þ ð Þ2H À1=2 ðdiv; CÞ Â H À1=2 ðdiv; CÞ, see [21] for the definition and the properties of the trace operators in H loc ðcurl; X e Þ. For non-smooth domains, see [7,8].
The exterior Calderón operator or admittance operator, C e , is defined as the mapping of the tangential component of the scattered electric field to the tangential component of the scattered magnetic field on the boundary of X [9]. We use the solution of a specific exterior problem to make the definition precise. Fig. 1 Typical geometry of the scattering problem in this paper. The domain X, its boundary C and the exterior X e 4 We use scaled electric and magnetic fields, i.e., the SI-unit fields E SI and H SI are related to the fields E and H used in this paper by where the permittivity and permeability of vacuum are denoted 0 and l 0 , respectively, and the relative permittivity and permeability of the exterior material are denoted and l, respectively. 5 Some authors [14] use c t for c and also use c T ¼ Àm Â c.
The following theorem represents the solution to Problem (E): Theorem 1 Let E s and H s be the solution of Problem (E). Then the fields satisfy the integral representations where the scalar Green function is The proof of this theorem is found in e.g., [14]. The second (lower) term of the integral representation, i.e., when x 2 X, is usually called the extinction part of the integral representation.
We notice that the exterior Calderón operator C e is uniquely defined for all m 2 H À1=2 ðdiv; CÞ, since Problem (E) has a unique solution in H loc ðcurl; X e Þ Â H loc ðcurl; X e Þ for any m 2 H À1=2 ðdiv; CÞ. Details on the space H À1=2 ðdiv; CÞ and its dual space H À1=2 ðcurl; CÞ are given in [9] and [20].

Theorem 2
The exterior Calderón operator defined in Definition 1 has the following properties [9]: 1. Positivity: where dS denotes the surface measure of C, and the star denotes the complex conjugation.
From Item 2 we conclude that the norm of the exterior Calderón operator satisfies kC e k H À1=2 ðdiv;CÞ ! 1, and also that the constants in Item 3 can be chosen as h C ¼ 1=kC e k H À1=2 ðdiv;CÞ and H C ¼ kC e k H À1=2 ðdiv;CÞ . Notice, that if we define the exterior Calderón operator with an extra imaginary unit (i), the exterior Calderón operator becomes its own inverse, i.e., C e : m7 !cðiH s Þ. This is a correction for the p=2 phase shift between the fields.

Integral equation approach
The results in Theorem 1 can be used to put the exterior Calderón operator in a surface integral equation setting. The following theorem is important for the analysis in this paper and proved in [14,Th. 5.52] (important results are also found in [10,12,26]): Theorem 3 Let Q be a bounded domain such that C Q.
1. Define the operators e L; f M : H À1=2 ðdiv; CÞ ! Hðcurl; QÞ, by These operators are well defined and bounded from the space H À1=2 ðdiv; CÞ into the space Hðcurl; QÞ. 2. For f 2 H À1=2 ðdiv; CÞ, the fields F ¼ f Mf and r Â F ¼ e Lf satisfy cðFÞj þ ÀcðFÞj À ¼ f ; cðr Â FÞj þ Àcðr Â FÞj À ¼ 0: The notation j AE refers to the trace of the field taken from the outside ðþÞ or the inside ðÀÞ of C, respectively. In particular, F 2 C 1 ðR 3 nCÞ, and F satisfies r Â r Â F ð ÞÀ k 2 F ¼ 0 in R 3 nC. Furthermore, the functions F and r Â F satisfy one of the two Silver-Müller radiation conditions uniformly w.r.t.x. 3. The traces L and M defined by are bounded from H À1=2 ðdiv; CÞ into itself. 4. For f 2 H À1=2 ðdiv; CÞ, the fields F ¼ f Mf and r Â F ¼ e Lf have traces 5. The operator L is the sum L ¼ b I þ K of an isomorphism b I from H À1=2 ðdiv; CÞ onto itself and a compact operator K. 6. The operator e L can be written as where the scalar single layer potential operator S is defined as where the surface integral is interpreted as a generalized integral (punctured surface by a circle). The corresponding vector-valued operator S is denoted by which is interpreted as the operator S applied to each Cartesian component of the tangential vector field f .

Theorem 4
The exterior Calderón operator satisfies Proof From the second representation in Theorem 1, we get by letting m ¼ cðE s Þ and C e ðmÞ ¼ cðH s Þ, We intend to take the trace c of this equation. In this limit process, the left-hand side becomes À 1 2 C e ðmÞ þ MC e ðmÞ, by the result of Theorem 3. This result holds, irrespectively from which side the limit is taken. The right-hand side has the limit and the result of the theorem follows. h

Generalized harmonics
The vector spherical harmonics constitute a well-established and important tool for the expansion of tangential vector fields on a spherical surface [17]. The main motivation behind this section is to generalize this tool to include also non-spherical surfaces.
We start this section by a review of two introduced differential operators that act on scalars and vectors, respectively. For simplicity, we assume that the surface C is simply connected. The eigenfunctions of these operators provide bases for L 2 ðCÞ and TL 2 ðCÞ, respectively. They are well suited for expansion of the traces of solutions to the Maxwell equations. The spherical surface case yields the well known vector spherical harmonics, see Appendix 1.
The scalar Laplace-Beltrami operator D C on C acting on a scalar field f is defined as The four intrinsic surface differential operators, div C ; curl C ; grad C ; curl C are defined in [4,9,21,25]. The vector Laplace-Beltrami operator D C on C acting on a tangential vector field f is defined as The scalar Laplace-Beltrami operator has a countable set of eigenfunctions in L 2 ðCÞ, which we denote fY n g 1 n¼1 , and they satisfy, see [21] ÀD The eigenvalues are all real, positive, and the only possible accumulation point of the eigenvalues is at infinity [16,21]. We order the eigenvalues as k 1 k 2 . . ., and normalizing the eigenfunctions fY n g 1 we obtain an orthonormal basis in L 2 ðCÞ, where, as above, a star Ã denotes complex conjugation. Notice that the eigenvalues are scaled with the wave number k 2 in order to have a dimensionless quantity, and moreover that the functions Y n have dimension inverse length, i.e., ½m À1 .
The following lemma is easily verified in view of the definitions of the scalar and vector Laplace-Beltrami operators.
Lemma 1 If f satisfies By the use of this lemma, we can construct a set of eigenfunctions to the vector Laplace-Beltrami operator. In the sequel, unless otherwise stated, we will consider that s ¼ 1; 2 and n; n 0 2 N ¼ f1; 2; 3; . . .g.

Definition 2
The vector generalized harmonics are defined as These functions have dimension inverse length, i.e., ½m À1 .
Remark 1 Note that Y 1n and Y 2n are eigenfunctions to the curl C curl C and Àgrad C div C operators, respectively. We also observe that Y 1n belongs to the kernel of the Àgrad C div C operator, and that Y 2n belongs to the kernel of the curl C curl C operator. Note also that for a simply-connected surface C, there is no eigenvalue k ¼ 0, see the end of proof of Lemma 2.
The following lemma proves that the set fY sn ; s ¼ 1; 2; n ¼ 1; 2; . . .g is an orthonormal system on TL 2 ðCÞ: The vector functions Y 1n and Y 2n defined in Definition 2 constitute an orthonormal basis on TL 2 ðCÞ, i.e., Z The vector functions satisfym where the dual index s is 1 ¼ 2 and 2 ¼ 1. Moreover, Proof We start by noticing that Y 1n and Y 2n 0 both are tangential to C, by the definition of the operators curl C and grad C . Equations (5), (6), (7), the relations hcurl C u; wi TL 2 ðCÞ ¼ hu; curl C wi L 2 ðCÞ , hdiv C u; /i L 2 ðCÞ ¼ Àhu; grad C /i TL 2 ðCÞ , together with curl C grad C / ¼ 0, and The final statements are easily proven by The completeness of the set of vector generalized harmonics fY 1n ; Y 2n g 1 n¼1 can be proved by investigating which f satisfies If this statement implies f ¼ 0, the set of vector generalized harmonics will be dense in TL 2 ðCÞ. We start with s ¼ 1, and get From the completeness of the generalized harmonics Y n (see, e.g., [21]), i.e., from the fact that hg; Y n i ¼ 0, 8n 2 N renders g ¼ 0, we obtain that curl C f ¼ 0. In the above, as well in the following relation, the brackets ðhÁ; ÁiÞ denote the suitable inner product or the appropriate duality pairing between the involved function spaces. We continue with s ¼ 2.
Again, the completeness of the generalized harmonics Y n implies div C f ¼ 0. However, a function f , which satisfies curl C f ¼ div C f ¼ 0 on a simply connected surface C, is zero [21, p. 206], and the lemma is proved. h 4 Trace spaces and the exterior Calderó n matrix

Spectral characterization of trace spaces
We redefine (in the spirit of [21]) the pertinent function spaces used frequently in this paper in terms of the orthogonal bases Y n and Y sn . The generalized Fourier series of a function f is f ¼ X n a n Y n ; a n ¼ hf ; Y n i L 2 ðCÞ ; where convergence is in the L 2 ðCÞ norm (defined below). The space L 2 ðCÞ is characterized as equipped with the norm kf k 2 L 2 ðCÞ ¼ X n a n j j 2 ; and the space H s ðCÞ is characterized as Similarly, the generalized Fourier series of a tangential vector function f is where convergence is in the TL 2 ðCÞ norm. The space TL 2 ðCÞ is characterized as and the space TH s ðCÞ is characterized as Remark 2 In [21] is this norm defined as kf k 2 TH s ðCÞ ¼ X sn k n ð Þ s a sn j j 2 : which is equivalent with (9) as long as the smallest eigenvalue is strictly positive.
The operations of curl C and div C imply, using Lemma 2, Note that only one of Y 1n and Y 2n survives the respective differentiation. This motivates the following redefinition of the involved spaces in terms of the corresponding suitable norms.
We also employ the weighted space ' À1=2 ðdivÞ defined by We notice that the spaces ' À1=2 ðdivÞ and H À1=2 ðdiv; CÞ are equivalent in the sense that f 2 H À1=2 ðdiv; CÞ if and only if its Fourier coefficients a sn 2 ' À1=2 ðdivÞ. We have the following Parseval type of identity Proof The proof follows from the construction of the vector generalized harmonics in Definition 2 and Lemma 2. h

The exterior Calderó n matrix
For simplicity, we assume that the surface C is simply connected. 7 Any m 2 H À1=2 ðdiv; CÞ \ TL 2 ðCÞ has a convergent Fourier expansion in terms of Y sn , i.e., Using Riesz representation, any m 2 H À1=2 ðdiv; CÞ has a generalized Fourier expansion in terms of the same basis as (10), where e sn 2 ' À1=2 ðdivÞ.
With the solution of Problem (E), the image of the exterior Calderón map C e ðmÞ 2 H À1=2 ðdiv; CÞ has an expansion and h sn 2 ' À1=2 ðdivÞ. Note the bar over the index s, which denotes the dual index in s (1 ¼ 2 and 2 ¼ 1), and the presence of an extra factor of i. The reason for this choice is that the expansion coefficients of the magnetic surface field then has a simple relation to the corresponding coefficients of the electric surface field. (10) is a Helmholtz-Hodge decomposition of the elements m in H À1=2 ðdiv; CÞ (and similarly of H À1=2 ðcurl; CÞ) and that the L 2 -projection can be interpreted as a duality pairing between H À1=2 ðdiv; CÞ and H À1=2 ðcurl; CÞ, see Remark 3.

Remark 4 We note that the expansion in
The mapping ' À1=2 ðdivÞ 3 e sn 7 !h sn 2 ' À1=2 ðdivÞ is a realization of the exterior Calderón operator. To every set of coefficients e sn there exists a unique set of coefficients h sn , and this association defines a linear relation between e sn 7 !h sn manifested by a matrix C (the exterior Calderón matrix) and The explicit form of the matrix is It is not hard to show that the exterior Calderón matrix C is invertible in ' À1=2 ðdivÞ.

Lemma 5
The exterior Calderón matrix C sn;s 0 n 0 ¼ ÀihC e ðY s 0 n 0 Þ; Y sn i TL 2 ðCÞ defined by (12) and (13) satisfies X s 00 n 00 C sn;s 00 n 00 C s 00 n 00 ;s 0 n 0 ¼ d ss 0 d nn 0 ; and its inverse is C À1 sn;s 0 n 0 ¼ C sn;s 0 n 0 : Proof The lemma is a consequence of ðC e Þ 2 ¼ ÀI on H À1=2 ðdiv; CÞ, the expansions in (10), (11), and the map (12). We have m ¼ ÀC e C e ðmÞ ð Þ; 8m 2 H À1=2 ðdiv; CÞ; or due to the continuity of the exterior Calderón operator X sn e sn Y sn ¼ Ài C sn;s 0 n 0 e s 0 n 0 C s 00 n 00 ;sn Y s 00 n 00 ¼ X sn X s 0 n 0 X s 00 n 00 C s 00 n 00 ;s 0 n 0 e s 0 n 0 C sn;s 00 n 00 Y sn ; since by (11) and (12) C e Y sn ð Þ ¼ i X s 00 n 00 C s 00 n 00 ;sn Y s 00 n 00 : Orthogonality then implies e sn ¼ X s 0 n 0 X s 00 n 00 C s 00 n 00 ;s 0 n 0 C sn;s 00 n 00 e s 0 n 0 ; SN Partial Differential Equations and Applications or, since the e sn are arbitrary X s 00 n 00 C sn;s 00 n 00 C s 00 n 00 ;s 0 n 0 ¼ X s 00 n 00 C s 00 n 00 ;s 0 n 0 C sn;s 00 n 00 ¼ d s;s 0 d n;n 0 : which ends the proof. h Moreover, we have Insert the expansions of m and C e ðmÞ, see (10) and (11). We obtain where we usedm Â Y s 0 n 0 ¼ ðÀ1Þ s 0 þ1 Y s 0 n 0 , see (8)  which proves the lemma. h Theorem 5 The norm of the exterior Calderón operator in H À1=2 ðdiv; CÞ is determined by the square root of the largest eigenvalue of the Hermitian matrix P ¼ D À1=2 C y D À1 CD À1=2 , i.e., the matrix P sn;s 0 n 0 ¼ X s 00 n 00 1 þ k n ð Þ Às=2þ3=4 C Ã s 00 n 00 ;sn 1 þ k n 00 ð Þ Às 00 þ3=2 C s 00 n 00 ;s 0 n 0 1 þ k n 0 ð Þ Às 0 =2þ3=4 ; where the diagonal matrix D is Proof The norms of the trace of the scattered electric and magnetic field are kcðE s Þk 2 H À1=2 ðdiv;CÞ ¼ X sn 1 þ k n ð Þ sÀ3=2 e sn j j 2 ; and kcðH s Þk 2 H À1=2 ðdiv;CÞ ¼ or in short-hand matrix notation kcðE s Þk 2 H À1=2 ðdiv;CÞ ¼ e y De; kcðH s Þk 2 H À1=2 ðdiv;CÞ ¼ h y D À1 h; where e and h are the column vectors of the coefficients e sn and h sn , respectively, and the matrix D is defined above. The Hermitian conjugate of these column vectors are denoted e y and h y . The norm of the exterior Calderón operator in H À1=2 ðdiv; CÞ can then be formed, viz.
This is a quadratic form and the largest eigenvalue of D À1=2 C y D À1 CD À1=2 determines the norm. h

Calculation of the exterior Calderó n matrix
The goal now is to find an explicit representation of the exterior Calderón matrix C sn;s 0 n 0 in terms of the geometry of the surface C. A number of lemmata and propositions are involved. Denote by S r the sphere of radius r centered at the origin, see Fig. 2. The restriction of cð f Mf Þ to S r defines an operator A r : H À1=2 ðdiv; CÞ ! H À1=2 ðdiv; S r Þ. The explicit expression of the operator is, for f 2 H À1=2 ðdiv; CÞ where the radius 0\r\R, R ¼ min x 0 2C jx 0 j. Define the radius a 2 ð0; RÞ such that the functions w l ðkaÞ 6 ¼ 0 and w 0 l ðkaÞ 6 ¼ 0 for all l ¼ 1; 2; . . ., where w l ðzÞ are the Riccati-Bessel functions [17,22]. This is always possible for small enough ka [ 0.
Proof The kernel of the operator A a is continuous (analytic in the variable x) and hence A a is compact. The operator is injective if we can prove that To accomplish this, define SN Partial Differential Equations and Applications By assumption, cðFÞ ¼ 0 on S a (the same limit from both sides). We proceed by proving that the only f that satisfies this condition is f ¼ 0.
Let B(a) denote the ball, centered at the origin, of radius a, see Fig. 2. The function FðxÞ satisfies, see Theorem 3 r Â ðr Â FðxÞÞ À k 2 FðxÞ ¼ 0; x 2 R 3 nC; therefore also in the ball B(a). Inside the ball B(a), the field FðxÞ has an expansion in regular spherical vector waves w sn kxÞ, defined by w 1n ðkxÞ ¼ xj l ðkxÞY 1n ðxjÞ where j l ðkxÞ is the spherical Bessel function of the first kind [23], and Y sn ðxÞ are vector harmonics for the sphere (vector spherical harmonics), see Appendix 1. Due to orthogonality of the vector spherical harmonics, and since a is chosen such that w l ðkaÞ 6 ¼ 0 and w 0 l ðkaÞ 6 ¼ 0 for all l ¼ 1; 2; . . ., the expansion coefficients of this expansion are all zero. Therefore, the interior boundary value problem has a unique solution FðxÞ ¼ 0, x 2 BðaÞ. By analyticity, FðxÞ ¼ 0 for all x 2 X [24]. As a consequence, the traces cðFÞj À ¼ 0 and cðr Â FÞj À ¼ 0. By Theorem 3, we also conclude that cðr Â FÞj þ ¼ 0.
As a function of x 2 X e , r Â FðxÞ satisfies the correct radiation conditions at infinity and cðr Â FÞj þ ¼ 0 on C. Due to unique solvability of the exterior problem (Problem (E)), r Â FðxÞ ¼ 0 in X e . Since F ¼ k À2 r Â ðr Â FÞ, FðxÞ ¼ 0 in X e , and, consequently, cðFÞj þ ¼ 0. Finally, the jump condition on the trace of F shows, see Theorem 3 This proves the injectivity of the operator A a .ν Γ Sa B(a) aν Fig. 2 The spherical surface S a and the domain X In order to prove that the range is dense, we define the adjoint operator A y a : H À1=2 ðcurl; S a Þ ! H À1=2 ðcurl; CÞ of A a w.r.t. to the dual spaces ðH À1=2 ðdiv; CÞ; H À1=2 ðdiv; S a ÞÞ. The explicit form of the adjoint operator is where (use rgðk; x À x 0 j jÞ¼ Àr 0 gðk; x À x 0 j jÞ) We now prove that A y a is injective, i.e., B is injective, namely To this end assume that Bg ð ÞðxÞ ¼ 0, x 2 C, and similarly as above, define the function so that by assumption, cð e FÞj AE ¼ Bg ð ÞðxÞ ¼ 0 on C (same limit from both sides). The function e FðxÞ satisfies r Â ðr Â e FðxÞÞ À k 2 e FðxÞ ¼ 0; x 2 R 3 nS a : Moreover, the function satisfies the appropriate radiation condition at infinity and cð e FÞj þ ¼ 0 on C. By the uniqueness of the exterior scattering problem (Problem (E)), e FðxÞ ¼ 0, x 2 X e , and by analyticity, e F ¼ 0 also outside S a . As above, by Theorem 3, the curl of e F has a continuous tangential component at S a . The interior problem is uniquely solvable, since w l ðkaÞ 6 ¼ 0 and w 0 l ðkaÞ 6 ¼ 0 for all l ¼ 1; 2; . . ., which implies that e FðxÞ ¼ 0, x 2 BðaÞ. The tangential components of e FðxÞ have a jump discontinuity on S a , Theorem 3. This proves the injectivity of the operator B, and, consequently, that the operator A a has a dense range, since NðA y a Þ ¼ RðA a Þ ? [6, p. 241]. where the dimensionless matrix A sn;s 0 n 0 is defined as The bar over the index s denotes the dual index in s (1 ¼ 2 and 2 ¼ 1).
Here u sn ðkxÞ are the radiating spherical vector waves, defined by where h ð1Þ l ðkxÞ is the spherical Hankel function of the first kind [23], see also Appendix 1. The matrix A sn;s 0 n 0 plays a central role in the procedure of calculating the norm of the exterior Calderón operator and it deserves a thorough study. This is done in Proposition 1 and Theorem 6 below.
Proof The extinction part of Theorem 1 reads Introduce the Green dyadic for the electric field in free space [17] G e ðk; x À x 0 Þ ¼ I 3 þ 1 k 2 rr gðk; x À x 0 j jÞ¼ I 3 þ 1 k 2 r 0 r 0 gðk; x À x 0 j jÞ; where I 3 is the unit dyadic in R 3 . Consequently, the extinction part is In fact, the curl on G e ðk; x À x 0 Þ gives r Â G e ðk; x À x 0 Þ ¼ r Â I 3 gðk; x À x 0 j jÞ ð Þ , which verifies (20).
The Green dyadic for the electric field is [17, (7.24) on p. 370] where x \ (x [ ) is the position vector with the smallest (largest) distance to the origin, i.e., if x\x 0 then x \ ¼ x and x [ ¼ x 0 . The definition of the spherical vector waves is given in Appendix 1, and, as before, the star Ã denotes complex conjugate. This expansion is uniformly convergent in compact (bounded and closed) domains, provided x 6 ¼ x 0 in the domain [15,20]. Apply (21) to (20) for an x inside the inscribed sphere of C and use the dual property of the spherical vector waves, i.e., r Â w sn ðkxÞ ¼ kw sn ðkxÞ; r Â u sn ðkxÞ ¼ ku sn ðkxÞ: We get Orthogonality of the vector spherical harmonics on the inscribed sphere implies Insert the expansion of the field in their Fourier series, (10) and (11), and we obtain which is identical to the statement in the lemma. Integration by parts gives an alternative form of the matrix A sn;s 0 n 0 , see (18) and use Definition 2.
and A sn; is injective, where the matrix A sn;s 0 n 0 is defined in (18).
Proof We prove the proposition by showing X s 0 n 0 A sn;s 0 n 0 a s 0 n 0 ¼ 0; 8n; s ¼ 1; 2; SN Partial Differential Equations and Applications implies that a sn ¼ 0 for s ¼ 1; 2 and all n.
Multiply this relation with w Ã sn ðkxÞ, where x lies inside the inscribed sphere of the scatterer, and sum over s and n. We obtain, see (21) 1 ik Now consider the vector-valued function which is defined everywhere in R 3 nC. This function is, by definition, zero inside the inscribed sphere of the scatterer. By analyticity, the function AðxÞ ¼ 0 for all x 2 X [24]. As a consequence, the traces cðAÞj À ¼ 0 and cðr Â AÞj À ¼ 0.
The curl of AðxÞ is The trace of FðxÞ has a jump discontinuity on C, see Theorem 3 and consequently, by orthogonality of the vector generalized harmonics, a sn ¼ 0, which implies the injectivity of the mapping above. h To simplify the analysis in the theorem below, we introduce a special notation for the matrix with dual s indices. To this end, define the matrix A sn;s 0 n 0 ¼ A sn;s 0 n 0 ; Theorem 6 The exterior Calderón matrix C can be approximated by C a sn;s 0 n 0 ¼ X s 00 n 00 X s 000 n 000 ðaI þ A y AÞ À1 sn;s 00 n 00 A Ã s 000 n 000 ;s 00 n 00 A s 000 n 000 ;s 0 n 0 ; for adequately small a [ 0, where y denotes the Hermitian conjugated matrix. In short- Proof The expansion coefficients e sn and h sn are related by, see (17) X s 0 n 0 A sn;s 0 n 0 h s 0 n 0 ¼ X This equation consists of a countable set of linear equations, the solution of which may be used to express h sn in terms of e sn , thus providing a matrix form representation of the exterior Calderón operator in terms of the chosen basis of generalized harmonics. Assuming the invertibility of the matrix A sn;s 0 n 0 , we write the equation as sn;s 00 n 00 A s 00 n 00 ;s 0 n 0 e s 0 n 0 ; so that C e admits the matrix representation sn;s 00 n 00 A s 00 n 00 ;s 0 n 0 ; However, by the definition of the matrix operator A and the connection of the spherical vector waves u s;n with the Green dyadic for the electric field, see left-hand side of (20) and (15), we see that A, and therefore also A, is related to a compact operator; hence A is not expected, in general, to be invertible and, even if it were, it would lead to an ill-posed problem which could not provide a well defined numerical scheme.
We may, however, resort to a Tikhonov regularization approach of the solution of (23), which leads to a, well-suited for numerical approaches, approximation of the exterior Calderón operator. According to the theory of the Tikhonov regularization, see [16,Ch. 16], the regularized approximate solution of (23) is h a sn ¼ X s 00 n 00 X s 000 n 000 ðaI þ A y AÞ À1 sn;s 00 n 00 A Ã s 000 n 000 ;s 00 n 00 A s 000 n 000 ;s 0 n 0 e s 0 n 0 ; a [ 0; or in shorthand matrix notation h a ¼ ðaI þ A y AÞ À1 A y Ae, which leads to an approximation of C by C a , where C a :¼ X s 00 n 00 X s 000 n 000 ðaI þ A y AÞ À1 sn;s 00 n 00 A Ã s 000 n 000 ;s 00 n 00 A s 000 n 000 ;s 0 n 0 ; a [ 0; or in shorthand matrix notation C a ¼ ðaI þ A y AÞ À1 A y A. The invertibility of the matrix aI þ A y A is easily obtained by the Lax-Milgram Lemma, since the regularization term aI introduces coercivity into the problem and the numerical inversion can be performed in terms of a variational approach related to the minimization problem min z2' À1=2 ðdivÞ akzk 2 ' À1=2 ðdivÞ þ hA y A z; zi ' À1=2 ðdivÞ : h The behavior as a ! 0 follows the general case setting of [16,Chap. 16].

The finite dimensional problem
This section contains a generalization of the result presented in [13] for a spherical surface to a general surface C. Denote We define the orthogonal projection P N : H À1=2 ðdiv; CÞ ! H À1=2 ðdiv; CÞ where f 7 !f N ¼ P N f in the H À1=2 ðdiv; CÞ inner product.
The following proposition holds: h Remark 6 The analysis can be extended for the case of non-simply-connected surfaces C, by extending the proposed orthonormal basis with the finite-dimensional basis of the kernel of the Laplace-Beltrami operator on C, see [21, p. 206].

Connection to the transition matrix for a PEC obstacle
Scattering by a perfectly conducting obstacle (PEC) with bounding surface C is related to the exterior Calderón operator C e . This section develops and clarifies this connection. The transition matrix (T-matrix), T sn;s 0 n 0 , connects the expansion coefficients of the incident field E i , with sources in X e and the scattering E s in terms of the regular spherical vector waves, w sn ðkxÞ, and the radiating spherical vector waves, u sn ðkxÞ, respectively. The definition of the spherical vector waves is given in Appendix 1. Specifically, where the regular and radiating spherical vector waves, w sn and u sn , are defined in (16) and (19), respectively, see also Appendix 1, and where the expansion coefficients f sn and a sn are related as f sn ¼ X s 0 n 0 T sn;s 0 n 0 a s 0 n 0 : The expansion of the incident field is absolutely convergent, at least, inside the inscribed sphere of the PEC obstacle, 8 and the expansion of the scattered field converges, at least, outside the circumscribed sphere of the PEC obstacle. The transition matrix completely characterizes the scattering process.
The following theorem shows that when the exterior Calderón operator is known, the transition matrix for a PEC obstacle is obtained by some simple operations: Theorem 7 The transition matrix for a PEC obstacle, T sn;s 0 n 0 , with bounding surface C and the corresponding exterior Calderón matrix, C sn;s 0 n 0 , is: T sn;s 0 n 0 ¼ i X s 00 n 00 W sn;s 00 n 00 V s 0 n 0 ;s 00 n 00 þ V s 0 n 0 ;s 00 n 00 X s 000 n 000 C s 000 n 000 ;s 00 n 00 W sn;s 000 n 000 ( )

;
where the dimensionless matrices W sn;s 0 n 0 and V sn;s 0 n 0 are W sn;s 0 n 0 ¼ k Z C w Ã sn Á Y s 0 n 0 dS; V sn;s 0 n 0 ¼ k Z C cðw sn Þ Á Y Ã s 0 n 0 dS: Notice that W sn;s 0 n 0 and V sn;s 0 n 0 are related, i.e., V sn;s 0 n 0 ¼ ðÀ1Þ sþ1 W Ã sn;s 0 n 0 .
Proof For a given incident field E i , the boundary condition on the surface C is cðE i þ E s Þ ¼ 0, which implies cðE s Þ ¼ ÀcðE i Þ: The trace of the scattered magnetic field on C is 8 More precisely, the convergence is guaranteed inside the largest inscribable ball not including the sources of the incident field.
cðHÞ ¼ cðH i þ H s Þ ¼ cðH i Þ À C e ðcðE i ÞÞ: The expansion coefficients of the scattered electric field for a PEC surface, f sn , are [17, (9.3) on p. 481] Inserting the expansions of the incident fields, we obtain an explicit form of the transition matrix, viz.
T sn;s 0 n 0 ¼ k 2 Z C w Ã sn Á icðw s 0 n 0 Þ þ C e ðcðw s 0 n 0 ÞÞ È É dS; where we also used the explicit form of the trace of the incident magnetic and electric fields H i ðxÞ ¼ Ài X sn a sn w sn ðkxÞ; E i ðxÞ ¼ X sn a sn w sn ðkxÞ: The regular spherical vector wave cðw sn Þ has a Fourier series expansion in Y sn .
kcðw sn Þ ¼ X s 0 n 0 V sn;s 0 n 0 Y s 0 n 0 ; V sn;s 0 n 0 ¼ k and (14) yields C e Y sn ð Þ ¼ i X s 00 n 00 C s 00 n 00 ;sn Y s 00 n 00 : Combine these expansions kC e cðw sn Þ ð Þ¼i X s 0 n 0 ;s 00 n 00 V sn;s 0 n 0 C s 00 n 00 ;s 0 n 0 Y s 00 n 00 ¼ i X s 0 n 0 ;s 00 n 00 V sn;s 0 n 0 C s 00 n 00 ;s 0 n 0 Y s 00 n 00 : These expressions lead to T sn;s 0 n 0 ¼ ik X s 00 n 00 Z C w Ã sn Á V s 0 n 0 ;s 00 n 00 Y s 00 n 00 þ V s 0 n 0 ;s 00 n 00 X s 000 n 000 C s 000 n 000 ;s 00 n 00 Y s 000 n 000 ( )

dS:
If we denote W sn;s 0 n 0 ¼ k Z C w Ã sn Á Y s 0 n 0 dS; we get in matrix notation T sn;s 0 n 0 ¼ i X s 00 n 00 W sn;s 00 n 00 V s 0 n 0 ;s 00 n 00 þ V s 0 n 0 ;s 00 n 00 X s 000 n 000 C s 000 n 000 ;s 00 n 00 W sn;s 000 n 000 ( ) ; which proves the theorem. h 6 The spherical geometry-an explicit example The spherical geometry is well-known and, so far, the only known geometry, where we can test the theory analytically. In this section, we apply the results above to a sphere of radius r. The eigenvalues for the sphere are 9 k n ¼ lðl þ 1Þ=ðkrÞ 2 , and the vector spherical harmonics Y sn ðxÞ, see [17] and Appendix 1.

> < > :
where j l ðkxÞ is the spherical Bessel function of the first kind [23].