Electromagnetic Steklov eigenvalues: existence and distribution in the self-adjoint case

In previous works, it was suggested to use Steklov eigenvalues for Maxwell equations as target signature for nondestructive testing, and it was recognized that this eigenvalue problem cannot be reformulated as a standard eigenvalue problem for a compact operator. Consequently, a modified eigenvalue problem with the desired properties was proposed. We report that apart for a countable set of particular frequencies, the spectrum of the original self-adjoint eigenvalue problem consists of three disjoint parts: The essential spectrum consisting of the origin, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a suitable topological decomposition, a representation of the operator as block operator and Schur-factorizations. For each Schur-complement, the existence of an infinite sequence of eigenvalues is proven via an intermediate value technique. The modified eigenvalue problem arises as limit of one Schur-complement.

these issues [2,4,10,12,16,17]. There is a lot of freedom in the construction of such methods, and a number of different versions have been proposed. Another improvement is the introduction of an additional sensitivity parameter as e.g. in [14,16,17] to tune the dependence of the eigenvalues with respect to changes in the material parameters. All of these emerging eigenvalue problems can roughly be classified in two types: transmission eigenvalue problems and Steklov eigenvalue problems. Among the important questions on these eigenvalue problems are: • characterization of the essential spectrum (i.e. eigenvalue parameters for which the operator is not Fredholm), • characterization of the Fredholm set (i.e. eigenvalue parameters for which the operator is Fredholm) and non-empty resolvent sets (in each connected subset), • existence of "enough" eigenvalues (as minimal requirement to be a meaningful target signature), e.g. infinitely many, • continuity of the eigenvalues with respect to changes in the material parameters, • further qualitative properties of the eigenvalue distribution as e.g. eigenvalue free zones, accumulation points and estimates on the location of certain characteristic (e.g. the smallest) eigenvalues, • reliable computational methods.
A classical approach to obtain such analytical results is to transform the eigenvalue problem first to a stencil K −λI with a compact operator K . While this is a very convenient technique, it is not always applicable. Indeed, this is not possible if multiple accumulation points of the spectrum exist, which was observed in [12] for electromagnetic Steklov eigenvalues. Hence, the authors of Camaño et al. [12] introduced a modification, which led to an alternative problem, which admits the desired properties. In [22], the Fredholmness and the discreteness of the spectrum of the original electromagnetic Steklov problem were reported by means of the T-coercivity technique, and in [21] the existence and stability of eigenvalues of the modified electromagnetic Steklov problem in conductive media were proven.
In this article, we consider the original electromagnetic Steklov eigenvalue problem in the self-adjoint case. We report a complete description of the spectrum (see Proposition 7): The spectrum consists of three disjoint parts: The essential spectrum consisting of the point zero, an infinite sequence of positive eigenvalues that accumulate only at infinity and an infinite sequence of negative eigenvalues that accumulate only at zero. The analysis is based on a representation of the operator as block operator. For small/big enough eigenvalue parameter, the Schur-complements with respect to different components can be built. For each Schur-complement, the existence of an infinite sequence of eigenvalues is proved via a fixed point/intermediate value technique as e.g. used in [11]. Roughly speaking, the original electromagnetic Steklov problem is a coupled system of an eigenvalue problem for a compact operator and an eigenvalue problem for the inverse of a compact operator. As a side result, we also analyze the spectrum of the modified electromagnetic Steklov eigenvalue problem, see Sect. 7. We report that the spectrum of the modified eigenvalue problem consists of an infinite sequence of eigenvalues, which accu-mulate only at +∞. 1 Our analysis further reveals that the modified eigenvalue problem arises as asymptotic limit for λ → +∞ of one of the Schur-complements. This clarifies the relation between the original and the modified problem.
As the original problem contains two kinds of eigenvalue sequences, it carries more information than the modified problem and hence seems to be better suited for inverse applications. In practice, only a few eigenvalues can be identified and thus an increase of observable target signatures is of great value. For example, significant targets are the smallest negative eigenvalue and the smallest positive eigenvalue.
At last, we address the asymptotic limit for λ → 0 of the respective Schur-complement. This eigenvalue problem is of a similar type as the modified Steklov problems considered in [14]. It has three interesting properties: It is independent of the permeability, the eigenvalues scale with the power minus two of the frequency and the eigenvalues suffice a min-max characterization.
The remainder of this article is organized as follows: In Sect. 2, we recall some results from [12] and explain the origin of the Steklov eigenvalue problem. In Sect. 3, we set our notation and formulate our assumptions on the domain and the material parameters. We also recall some classical regularity, embedding and decomposition results, which will be essential for our analysis and adapt them to our setting. In Sect. 4, we introduce the considered electromagnetic Steklov eigenvalue problem and define the associated holomorphic operator function A X (·). We establish in Theorem 2 that the spectrum of A X (·) is real and that A X (λ) is Fredholm if and only if λ = 0. In Sect. 5, we analyze the spectrum in a neighborhood of zero. We report in Theorem 3 that there exists c 0 > 0 so that σ A X (·) ∩ (0, c 0 ) = ∅. We establish in Theorem 4 the existence of an infinite sequence of negative eigenvalues which accumulate at zero. In Sect. 6, we analyze the spectrum in a neighborhood of infinity. We report in Theorem 5 that there exists c ∞ > 0 so that σ A X (·) ∩ (−∞, −c ∞ ) = ∅. We establish in Theorem 6 the existence of an infinite sequence of positive eigenvalues, which accumulate at +∞. In Sect. 7, we collect our results in Proposition 7 and comment on the connection between the original and the modified electromagnetic Steklov eigenvalue problems.

Inverse scattering
In this section, we recall the discussion from Camaño et al. [12] to explain the relation between the Steklov eigenvalue problem and nondestructive inverse scattering methods. Let S 2 = {x ∈ R 3 : |x| = 1}. We consider a plane wave with direction of propagation d ∈ S 2 , polarization vector p ∈ R 3 \ {0} and frequency ω ∈ R \ {0}. Let ∈ L ∞ (R 3 ) be the relative permittivity such that inf x∈R 3 ( ) > 0 and ( ) ≥ 0. Let D ⊂ R 3 be a bounded Lipschitz domain such that = 1 on R 3 \ D. We consider the forward scattering problem to find u such that The scattered field u s has the following asymptotic expansion whereby we call u ∞ (x, d; p) the far-field pattern of u s for measurement directionx ∈ S 2 , incident direction d and polarization p. Let F : L 2 t (S 2 ) → L 2 t (S 2 ) be the far-field operator defined by We consider the following inverse problem. Let either B = D or B ⊂ R 3 be a bounded Lipschitz domain with D in its interior. Given the far-field pattern for allx, d, p we wish to compute approximations of Steklov eigenvalues (that we shall define shortly). To this end, we introduce the -independent auxiliary scattering problem to find u λ such that whereby λ is a real constant. Note the sign of λ herein is reversed compared to Camaño et al. [12]. The existence and uniqueness of solutions to (2) and (5) are established in the following manner. The problems are reformulated on a ball B R by means of the capacity operator. The injectivity is shown by testing with u = u, taking the imaginary part of the sesquilinear form and applying properties of the capacity operator. If the problem is Fredholm, then the bijectivity follows. The Fredholmness of the operators related to (2) and (5) for λ ≤ 0 can be shown by standard techniques. The case λ > 0 can be treated as in [22]. Let u λ,∞ be the far-field pattern of u s λ and let Consider a Herglotz wave function v g (x) := iω with Herglotz kernel g. Then, the weighted far-field pattern is the far-field pattern of the scattered part of the solution w of Analogously is the far-field pattern of the scattered part of the solution w λ of If (F − F λ )g = 0, then the far-field patterns agree w ∞ = w λ,∞ . Then, by Rellich's Lemma w = w λ in R 3 \ D. It follows that w is a nontrivial solution to curl curl −ω 2 w = 0 in B, The problem to find λ ∈ C and nontrivial w such thas (12) up to linear terms.

General setting
In this section, we set our notation and formulate assumptions on the domain and material parameters. We also recall necessary results from different literature and adapt them to our setting.

Functional analysis
Let (X, · X ), (Y, · Y ) be generic Banach spaces. We denote the space of all bounded linear operators from X to Y as L(X, Y ) with operator norm A L(X,Y ) := sup u∈X\{0} Au Y / u X , A ∈ L(X, Y ). In addition, we set L(X) := L(X, X). For A ∈ L(X, Y ), we denote A * ∈ L(Y, X) its adjoint operator defined through u, A * u X = Au, u Y for all u ∈ X, u ∈ Y . We call the space of compact operators as K (X, Y ) ⊂ L(X, Y ) and K (X) := K (X, X). We say that an operator A ∈ L(X) is coercive, if inf u∈X\{0} | Au, u X |/ u 2 X > 0. We say that A ∈ L(X) is weakly coercive, if there exists K ∈ K (X) so that A + K is coercive. Let

Lebesgue and Sobolev spaces
Let ⊂ R 3 be a bounded path connected open Lipschitz domain and ν the outer unit normal vector at ∂ . Let C ∞ 0 ( ) be the space of infinitely many times differentiable functions from to C with compact (closure of the) support in . We use standard notation for Lebesgue and Sobolev spaces L 2 ( ), L ∞ ( ), W 1,∞ ( ), H s ( ) defined on the domain and L 2 (∂ ), H s (∂ ) defined on the boundary ∂ . We recall the continuity of the trace operator tr ∈ L H s ( ), H s−1/2 (∂ ) for all s > 1/2. For a vector space X of scalarvalued functions, we denote its bold symbol as space of three-vector-valued functions X := X 3 = X × X × X, e.g.

Additional function spaces
Denote ∂ x i u the partial derivative of a function u with respect to the variable x i . Let For ∈ L ∞ ( ) 3x3 let div u := div( u). For a bounded Lipschitz domain , let ∇ ∂ , div ∂ and curl ∂ = ν × ∇ ∂ be the respective differential operators for functions defined on ∂ . We recall that for u ∈ L 2 ( ) with curl u ∈ L 2 ( ) the tangential trace is well defined and tr ν× u 2 H −1/2 (div ∂ ;∂ ) is bounded by a constant times u 2 L 2 ( ) + curl u 2 L 2 ( ) . Likewise for u ∈ L 2 ( ) with div u ∈ L 2 ( ) the normal trace tr ν· u ∈ H −1/2 (∂ ) is well defined and tr ν· u 2 H −1/2 (∂ ) is bounded by a constant times u 2 L 2 ( ) + div u 2 L 2 ( ) . Likewise for u ∈ L 2 ( ) with div u ∈ L 2 ( ) the normal trace tr ν· u ∈ H −1/2 (∂ ) is well defined and tr ν· u 2 and

Assumption on the domain and material parameters
Assumption 3.1 (Assumption on ) Let ∈ L ∞ ( ) 3x3 be a real, symmetric matrix function so that there exists c > 0 with c |ξ | 2 ≤ ξ H (x)ξ for all x ∈ and all ξ ∈ C 3 . We further assume that there exists a Lipschitz domainˆ ⊂ so that the closure ofˆ is compact in and | \ˆ equals the identity matrix I 3×3 ∈ C 3×3 .
Letˇ ⊂ be a Lipschitz domain so that the closure ofˇ is compact in and the closure ofˆ ⊂ˇ is compact inˇ . Let χ be infinitely many times differentiable, so that χ| \ˇ = 1 and χ|ˆ = 0.
To our knowledge, the most general todays available result on the unique continuation principle for Maxwell's equations is the one of Ball et al. [5]. It essentially requires and μ −1 to be piece-wise W 1,∞ .
We deduce the next lemma from Amrouche et al. [1].

Helmholtz decomposition on the boundary
We recall from Buffa, Costabel and Sheen [7, Theorem 5.5]: and denote the respective orthogonal projections by Recall div ∂ tr ν× ∈ L H(curl; for all z ∈ H 1 * (∂ ) and set From the construction of S, it follows S ∈ L H(curl; ), L 2 t (∂ ) and further

The electromagnetic Steklov eigenvalue problem
Let ω > 0 be fixed. For λ ∈ C, let A(λ) ∈ L H(curl, tr ν× ; ) be defined through The electromagnetic Steklov eigenvalue problem, which we investigate in this note, is to We note that the sign of λ herein is reversed compared to [12]. Let for all u, u ∈ H(curl, tr ν× ; ). It is straightforward to see that the norms induced by ·, · X and ·, · H(curl,tr ν× ; ) are equivalent. To analyze the operator A(λ), we introduce the following subspaces of H(curl, tr ν× ; ): We recall [26,Theorem 4.3 and Remark 4.4]: and dim K N ( ) = number of connected parts of ∂ − 1 < ∞. It holds Thus, tr ν· ∇u, 1 L 2 ( ) = 0 for each of the connected parts of ∂ }.
We continue with a decomposition of H(curl, tr ν× ; ), which is similar but different to Halla [22, in the following sense. There exist projections P V , 2a.
Step: Let u ∈ H(curl, tr ν× ; ). Note that due to div (u−P W 2 u) = 0 and Lemma 3.6 it hold tr ν· (u − P W 2 u) ∈ L 2 (∂ ) and tr ν· (u − P W 2 u), 1 L 2 ( ) = 0 for each of the connected parts of ∂ . Let w * ∈ H 1 * ( ) be the unique solution to Let P W 1 u := ∇w * . By construction of P W 1 and due to Lemma 3.6, it holds P W 1 ∈ L H(curl, tr ν× ; ) and ran P W 1 ⊂ W 1 . Let u ∈ W 1 . Then, P W 2 u = 0 and hence P W 1 u = u. Thus, P W 1 is a projection and ran P W 1 = W 1 . 2b.
Step: Let u ∈ H(curl, tr ν× ; ) and P V u := u − P W 1 u − P W 2 u. It follows P V ∈ L H(curl, tr ν× ; ) , P V u ∈ V and P V P V u = P V u. If u ∈ V , then P V u = u, and hence, ran P V = V . It follows further W 1 , W 2 ⊂ ker P V .

4.
Step: By means of the triangle inequality and a Young inequality, it holds.
On the other hand, due to the boundedness of the projections Thus, · X is equivalent to · X . Since · X is equivalent to · H(curl,tr ν× ; ) , · X is also equivalent to · H(curl,tr ν× ; ) .
Let us look at A(λ) in light of this substructure of H(curl, tr ν× ; ). To this end, we consider the space X := H(curl, tr ν× ; ), ·, · X as defined in (34).
It follows that P V , P W 1 and P W 1 are even orthogonal projections in X. Let further A X (·), A c , A , A tr ∈ L(X) be defined through A u, u X := u, u L 2 ( ) for all u, u ∈ X, A tr u, u X := tr ν× u, tr ν× u L 2 t (∂ ) for all u, u ∈ X.
From the definitions of V, W 1 and W 2 , we deduce that Proof The first statement follows from Theorem 3.2 and Corollary 3.4 of [22]. The second statement can be seen as in the proof of Corollary 3.3 of [22].
From (37) or (38), we recognize that any eigenfunction u ∈ X satisfies P W 2 u = w 2 = 0. Hence, to study the eigenvalues of A X (·) it suffices to study

Spectrum in the neighborhood of zero
First, we establish in Theorem 3 the absence of eigenvalues of A X (·) in (0, c) for sufficiently small c > 0. Later on in Theorem 4, we establish the existence of an infinite sequence of negative eigenvalues of A X (·) which accumulate at zero.

Spectrum right of zero
We will require in this section the following additional assumption.
and thence it holds For λ satisfying (40), we build the Schur-complement of (P V + P W 1 )A X (λ)| V ⊕W 1 with respect to P V u = v: It is straightforward to see, that for λ satisfying (40), λ is an eigenvalue to A X (·) if and only if λ is an eigenvalue to A W 1 (·). Hence, to study the eigenvalues of A X (·) in a neighborhood of zero, it completely suffices to study the eigenvalues of A W 1 (·) in a neighborhood of zero. For we deduce Let B tr ∈ L X, L 2 t (∂ ) : u → tr ν× u so that Proof A tr is self-adjoint and positive semi-definite due to (45) and hence so is P W 1 A tr | W 1 . P W 1 A tr | W 1 is weakly coercive due to Lemma 3.8 and curl w 1 = 0 for each w 1 ∈ W 1 . P W 1 A tr | W 1 is injective since w 1 ∈ W 1 ∩ ker(P W 1 A tr | W 1 ) implies w 1 ∈ W 2 and hence w 1 = 0. Since P W 1 A tr | W 1 is self-adjoint, positive semi-definite and bijective, it is already strictly positive definite.
Proof It is straightforward to see that H W 1 (λ) self-adjoint for λ ∈ R satisfying (40). The inverse triangle inequality, Lemma 5.2 and (43), (44) yield the claim. Proof For λ ∈ (0, c 0 ), we can build the Schur-complement A W 1 (λ) of A X (λ) with respect to P V u = v and A X (λ) is bijective if and only if A W 1 (λ) is so. It follows from the definition (42a) of A W 1 (λ) and Lemma 5.3 that A W 1 (λ) is strictly positive definite for λ ∈ (0, c 0 ) and hence bijective.

Spectrum left of zero
To study the eigenvalues of A W 1 (·) in (−c 0 , 0), we introduce We notice that λ ∈ (−c 0 , 0) is an eigenvalue of A W 1 (·), if and only if τ is an eigenvalue of A W 1 (·, λ) and τ = λ. We prove the existence of infinite eigenvalues of A W 1 (·) in (−c 0 , 0) by the fixed point technique outlined in [11]. Proof Due to Lemma 5.3 (P W 1 A tr | W 1 +H W 1 (λ)) −1/2 is well defined and self-adjoint. It holds dim W 1 = ∞ due to (32). The spectra of A W 1 (·, λ) and coincide. The latter is the pencil of a standard eigenvalue problem for a compact selfadjoint non-positive injective operator on an infinite-dimensional Hilbert space and respective properties follow. Proof Let (τ n (λ)) n∈N be as in Lemma 5.5. Let λ ∈ (−c 0 , 0). Let n 1 ∈ N be so that λ < τ n 1 (λ).

Spectrum in the neighborhood of infinity
First, we establish in Theorem 5 the absence of eigenvalues of A X (·) in the interval (−∞, −c) for sufficiently large c > 0. Later on in Theorem 6, we establish the existence of an infinite sequence of positive eigenvalues of A X (·) which accumulate at +∞.

The spectrum near negative infinity
We require the following additional assumption for Theorem 5.
Proof We are not aware of a direct appropriate reference for this lemma. Although we believe that the technique applied in this proof is common knowledge. We introduce mixed equations for u (and u λ ) as e.g. in [28] as follows. Letf ∈ X be so that f , u X = f, u L 2 ( ) for all u ∈ X. Due to u ∈ H(curl, tr 0 ν× ; ) and Assumption 3.2, it follows φ := ν × tr ν× μ −1 curl u ∈ L 2 t (∂ ). It holds φ λ := ν × tr ν× μ −1 curl u λ = λ tr ν× u λ ∈ L 2 t (∂ ) too. Integration by parts yields that (u, φ), and Proof Assume the contrary. Thus, there exists a sequence (λ n < 0) n∈N with lim n∈N λ n = −∞, so that A X (λ n ) is not bijective. Due Theorem 2 (λ n ) n∈N are eigenvalues of A X (·).
Hence, let (u n ∈ X) n∈N be a corresponding sequence of normalized eigenfunctions: A X (λ n )u n = 0 and u n X = 1 for each n ∈ N. It follows As already discussed at the end of Sect. 4, it holds u n ∈ V ⊕ W 1 for each n ∈ N. Denote E ∈ L X, L 2 ( ) the embedding operator and M ∈ L L 2 ( ) the multiplication operator with symbol . Thus, A = E * M E. Due to Lemma 3.8, there exist f ∈ L 2 ( ) and a subsequence (n(m)) m∈N so that lim m∈N Eu n(m) = f . Let u ∈ H(curl, tr 0 ν× ; ) be the solution to curl μ −1 curl u + u = f in . It follows from Lemma 6.2 and (51) that lim m∈N u n(m) = (ω 2 + 1)u in X. Since curl μ −1 curl u n(m) − ω 2 u n(m) = 0 in for each m ∈ N, it follows that curl μ −1 curl u − ω 2 u = 0 in as well. Due to Assumption 6.1 it holds u = 0, which is a contradiction to u n(m) X = 1 for each m ∈ N. The claim is proven.

The spectrum near positive infinity
is strictly positive definite due to Lemma 5.2. Hence, there exists c ∞ > 0 so that is coercive and thus bijective for each λ ∈ C with |λ| > c ∞ . (Since A is positive semi definite, it follows even that P W 1 (ω 2 A +λA tr )| W 1 is coercive for each λ ∈ C\R − 0 . However, we will not use this fact.) Hence for |λ| > c ∞ we build and study the Schur-complement of (P V + P W 1 )A X (λ)| V ⊕W 1 with respect to P W 1 u = w 1 : It is straightforward to see that for λ satisfying |λ| > c ∞ , λ is an eigenvalue to A X (·) if and only if λ is an eigenvalue to A V (·). Hence to study the eigenvalues of A X (·) in a neighborhood of infinity, it completely suffices to study the eigenvalues of A V (·) in a neighborhood of infinity. It will be more convenient to work with λ −1 instead of λ. Hence letÃ forλ ∈ C with |λ| < c −1 ∞ . Again, it is straightforward to see thatλ with 0 < |λ| < c −1 ∞ is an eigenvalue toÃ V (·) if and only ifλ −1 with |λ −1 | > c ∞ is an eigenvalue to A V (·). Thus, we study the eigenvalues ofÃ V (·) in the ball To this end, we introduce We note thatλ ∈ B c −1 ∞ is an eigenvalue ofÃ V (·), if and only ifτ is an eigenvalue ofÃ V (·,λ) andτ =λ ∈ B c −1 ∞ . We would like to proceed as in Sect. 5. OperatorK V (λ) is compact due Lemma 3.7. However, different to Sect. 5, Therefore, we introduce the abstract Lemma 6.3. Subsequently, we prove that the conditions of Lemma 6.3 are satisfied and the lemma can be employed for our particular application. We derive the results aimed at in Lemma 6.12 and consequently continue the analysis in the same manner as in Sect. 5. Lemma 6.3 Let Y be a separable Hilbert space. Let G ∈ L(Y ) be compact, self-adjoint and I + G be bijective. Let K ∈ L(Y ) be compact, self-adjoint, positive semi-definite and so that ker K = ker(K 1/2 (I + G)K 1/2 ) and dim(ker K ) ⊥ = ∞. Let P (ker K ) ⊥ be the orthogonal projection onto (ker K ) ⊥ and P (ker K ) ⊥ (I + G)| (ker K ) ⊥ be bijective.
Step: Since K 1/2 (I + G)K 1/2 is compact and self-adjoint and Y is separable with dim Y ≥ dim(ker K ) ⊥ = ∞ the Spectral Theorem for compact, self-adjoint operators yields: The spectrum of K 1/2 (I + G)K 1/2 consists of the essential spectrum {0} and an infinite sequence of eigenvalues (τ n ∈ R) n∈N (with multiplicity taken into account), lim n∈N τ n = 0 and there exists an orthonormal basis (y n ) n∈N of corresponding eigenelements. Due to dim(ker K ) ⊥ = ∞, there exists an infinite index set M ⊂ N so that τ m = 0 for each m ∈ M. 3.
Step: It remains to prove that all (τ m ) m∈M apart from a finite set are positive. To this end, we apply a technique which is inspired by [25,Sect. 3]. Let Y := span{y m : m ∈ M} cl = (ker K 1/2 (I + G)K 1/2 ) ⊥ = (ker K ) ⊥ and denote PỸ the orthogonal projection ontoỸ . We note that for each y ∈ Y , y 0 ∈ ker K it holds Thus, ran K 1/2 ⊂ (ker K ) ⊥ =Ỹ and so (τ I + K ) 1/2Ỹ ⊂Ỹ . Let G = G + − G − so that G + and G − are compact, self-adjoint and positive semi-definite, i.e. a decomposition of G in the positive and the negative part. For τ > 0, we compute By means of the Spectral Theorem for compact, self-adjoint operators, we deduce that Proof Letλ ∈ [0, c −1 ∞ ).K V (λ) is compact due Lemma 3.7. It follows from the definition of Hence Letλ It follows w 1 = 0 and hence w 1 , w 1 L 2 ( ) > 0. Since On the other hand: If B tr v = 0, then alsoK V (λ)v = 0 due to the definition ofK V (λ). Thus kerK V (λ) = ker B tr for each λ ∈ (0, c −1 ∞ ).

Lemma 6.5 Let Assumptions
Proof Let P ∈ L L 2 t (∂ ) be the L 2 t (∂ )-orthogonal projection onto the closure of ran B tr | W 1 . It follows from the definition ofK V (0) thatK V (0) = B * tr (I − P)B tr . The claim is proven, if we show that ran B tr | W 1 = curl ∂ H 1 (∂ ). It follows from the definition of W 1 that ran B tr | W 1 ⊂ curl ∂ H 1 (∂ ). Let φ ∈ curl ∂ H 1 (∂ ) and ψ ∈ H 1 (∂ ) so that φ = curl ∂ ψ = ν ×∇ ∂ ψ. Letw ∈ H 1 ( ) solve div ∇w = 0 in and trw = ψ at ∂ . With Proof Let (f n ) n∈N be an orthonormal basis of ∇ ∂ H 1 (∂ ) ⊂ L 2 t (∂ ). Let u n ∈ H(curl; ) be so that tr ν× u n = f n . Hence u n ∈ X. It follows Thus if N n=1 c n (P V u n + ker P ∇ ∂ B tr | V ) would be a nontrivial linear combination of zero in V /(ker P ∇ ∂ B tr | V ), then N n=1 c n f n would be a nontrivial linear combination of zero in ∇ ∂ H 1 (∂ ). Hence, dim V /(ker P ∇ ∂ B tr | V ) = +∞. Since ker B tr ⊂ ker P ∇ ∂ B tr , it follows dim V /(ker B tr | V ) ≥ dim V /(ker P ∇ ∂ B tr | V ) and thus the dimension of dim V /(ker B tr | V is infinite too. The claim follows from dim V /Z = dim Z ⊥ V for any closed subspace Z ⊂ V . We require the following additional assumption for Lemma 6.8. Assumption 6.7 (ω 2 is no "Dirichlet" eigenvalue) Let Z 1 := {z ∈ V : B tr z = 0} = H(curl, div 0 , tr 0 ν× , tr 0 ν· ; ) and denote P Z 1 the X-orthogonal projection onto Z 1 . The operator is bijective.
We require the following additional assumption for Lemma 6.10. Assumption 6.9 (ω 2 is no "hybrid" eigenvalue) Let and denote P Z 2 the X-orthogonal projection onto Z 2 . The operator is bijective. Lemma 6.10 Let Assumptions 3.1-3.3, 5.1 and 6.9 hold true. Thence, It follows P ∇ ∂ B tr z = 0 due to ker K V (0) 1/2 = ker K V (0) and Lemma 6.5. Due to the definitions of z and Z 2 , z ∈ Z 2 solves It follows from Assumption 6.9 that z = 0. Thus, v ∈ ker K V (0) 1/2 = ker K V (0).
We require the following additional assumption for Lemma 6.12.
Proof We note thatÃ V (·,λ) and V (·,λ) have the very same spectral properties. We aim to apply Lemma 6.3 to with G defined as in (60). G is compact due to Lemmas 3.8 and 3.7. Since P V (A c − ω 2 A )| V −1 and the identity are self-adjoint, the self-adjointness of G follows from (60).
Proof Proceed as in the proof of Theorem 4.

Conclusion
We conclude with a summary of Theorems 2-6 and some remarks on assumptions and the relation to the modified electromagnetic Steklov eigenvalue considered in [12,22].

Main result
We formulate the individual results of the previous sections in the following proposition. Proof Follows from Theorems 2-6.

Remarks to the assumptions
The condition in Assumptions 3.1 and 3.2 that μ and equal the identity matrix in a neighborhood of the boundary is used to obtain extra regularity of traces. If this extra regularity can be derived by other means, then the mentioned assumption becomes obsolete.
Each of the Assumptions 5.1, 6.1, 6.7, 6.9, 6.11 can be formulated in the following manner: Y is a Hilbert space, A ∈ L(Y ) is weakly coercive, K (·): ⊂ C → K (Y ) is holomorphic and it is imposed that A − K (ω 2 ) is bijective. Consequently, for fixed domain and fixed material parameters μ −1 , there exists only a countable set of frequencies ω for which the Assumptions 5.1, 6.1, 6.7, 6.9, 6.11 are not satisfied (see e.g. [24, Proposition A.8.4]).

Modified electromagnetic Steklov eigenvalues
The modified electromagnetic Steklov eigenvalue problem considered in [22] is to find (λ, u) ∈ C × H(curl; ) \ {0} so that μ −1 curl u, curl u L 2 ( ) − ω 2 u, u L 2 ( ) − λ Su, Su L 2 t (∂ ) = 0 (62) for all u ∈ H(curl; ) (with S defined as in (24)). It can easily be seen that the eigenvalue problem decouples with respect to the decomposition H(curl; ) = H(curl, div 0 , tr 0 ν· ; ) ⊕ ∇H 1 ( ). Thus, the eigenvalue problem can be reformulated to find (λ, u) ∈ C × H(curl, div 0 , tr 0 ν· ; )\{0} so that 0 = μ −1 curl u, curl u L 2 ( ) − ω 2 u, u L 2 ( ) − λ Su, Su L 2 t (∂ ) = μ −1 curl u, curl u L 2 ( ) − ω 2 u, u L 2 ( ) − λ P ∇ ∂ tr ν× u, tr ν× u L 2 for all u ∈ H(curl, div 0 , tr 0 ν· ; ). Thence if the respective assumptions are satisfied, Lemma 6.12 yields that the spectrum consists of an infinite sequence of eigenvalues (λ n ) n∈N which accumulate only at +∞. A similar existence result has been reported in [12,Theorem 3.6]. Though it seems to us that the proof of [12, Theorem 3.6] requires dim(ker T) ⊥ = ∞ which the authors don't elaborate on. The former observation admits to interpret the modified electromagnetic Steklov eigenvalue problem as asymptotic limit of the original electromagnetic Steklov eigenvalue problem for large eigenvalue parameter λ. We have seen that (at least in the self-adjoint case) the original electromagnetic Steklov eigenvalue problem yields two kinds of spectra. Contrary the modified electromagnetic Steklov eigenvalue problem yields only one kind of spectrum. This suggests that for inverse scattering applications the original version is more advantageous than the modified version, because it contains more information, though the approximation of the modified eigenvalue problem is better understood than for the original version [22].

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Data availability
Data sharing was not applicable to this article as no datasets were generated or analyzed during the current study.