The quasi-static plasmonic problem for polyhedra

We characterize the essential spectrum of the plasmonic problem for polyhedra in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra and permittivities $\epsilon<- 1$. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann--Poincar\'e operator. We therefore study the spectral structure of the the double layer potential for polyhedral cones and polyhedra.


Background.
Let Ω ⊂ R 3 be an open simply connected bounded polyhedron (with flat faces and straight edges), understood as an inclusion into infinite space with relative permittivity ∈ C, Re < 0. For a given function (or distribution) g on ∂Ω, the quasi-static plasmonic problem seeks a potential U : R 3 → C, which is harmonic in Ω and its exterior, and satisfies Here Tr ± U and ∂ ∂n U ± denote the interior/exterior traces and limiting outward normal derivatives of U on ∂Ω. A value of for which there is a non-zero solution U of the plasmonic problem with g = 0 is a plasmonic eigenvalue; the corresponding eigenfield ∇U is a static plasmon associated with the permittivity .
If Ω is Lipschitz and U is assumed to be of finite energy, then any plasmonic eigenvalue must satisfy that < 0, since Green's formula implies that R 3 \Ω |∇U | 2 dx = − Ω |∇U | 2 dx. Plasmonic problems, where Re < 0, appear as quasi-static approximations of electrodynamical problems where the scatterer is much smaller than the wavelength of the scattered electromagnetic wave, see [2] and [19,Section 8]. If instead > 0, or if Im = 0 and Re ε > 0, then the described problem is an ordinary electrostatic or quasi-static problem, which is thoroughly studied and mostly very well understood.
We will use layer potential operators to interpret and analyze the plasmonic problem of Ω as a spectral problem. Given a charge u on ∂Ω, the corresponding single layer potential of −∆ is given by Su(x) = 1 4π ∂Ω S(x, y)u(y) dS(y) = 1 4π ∂Ω u(y) |x − y| dS(y), where dS denotes the standard surface measure on ∂Ω. The Neumann-Poincaré operator on ∂Ω, the direct value of the double layer potential of u, is defined by where n y is the outward normal vector at (almost every) y ∈ ∂Ω. Inserting the ansatz U = Su into the plasmonic problem yields the equation where the adjoint K * has been formed with respect to the L 2 (∂Ω)-pairing. Note that Re < 0 if and only if |λ| < 1/2. For justification of the connection between the plasmonic problem and the spectral theory of K in L p , Sobolev, and Hardy space settings, see for example [1,13,18,20]. For a treatment that includes classes of non-Lipschitz domains, see [22]. For smooth domains, the spectrum of the plasmonic problem consists solely of a sequence of eigenvalues, which for 3D domains is governed by a Weyl law [3,37]. For domains with singularities, the plasmonic problem also exhibits essential spectrum (here interpreted via the connection with the Neumann-Poincaré operator). To exemplify this, we recall that for a curvilinear polygon Ω in 2D, with interior angles β 1 , . . . , β J , the spectral picture of the analogous 2D Neumann-Poincaré operator is very well understood [5,7,33,40,41]. As is typical for domains with singularities, the situation is highly dependent on the choice of function space. On the Sobolev space H 1/2 (∂Ω), the most physically meaningful choice, there is a self-adjoint realization of K : H 1/2 (∂Ω) → H 1/2 (∂Ω), and the essential spectrum is absolutely continuous and given by (1.2) σ ess K, H 1/2 (∂Ω) = x ∈ R : |x| ≤ max 1≤j≤J |1 − β j /π| 2 .
For polyhedra Ω ⊂ R 3 , the study of the (essential) spectral radius of K : X → X and the invertibility of K + 1/2 : X → X is a topic of very rich history; a vast variety of function spaces X on ∂Ω have been considered. The invertibility of K + 1/2 reflects the possibility of solving the Dirichlet problem in Ω with boundary data from X. We refer to [51] for an extensive survey, choosing here to only summarize the state of the art as it relates to the plasmonic problem.
Rathsfeld [43] proved that K + 1/2 (appropriately modified at the edges) is invertible on the space C(∂Ω) of continuous functions on ∂Ω, for arbitrary polyhedra Ω. To prove his result, Rathsfeld estimated the spectral radius on polyhedral cones using Mellin techniquesnote well the important correction that was made to this analysis in [44]. Elschner [12] refined the analysis further and proved that the essential spectral radius of K is less than 1/2 for a range of weighted Sobolev spaces on Lipschitz polyhedra ∂Ω; in Lemma 4.12 we will recall some important details of Elschner's study. Grachev and Maz'ya independently obtained the same result as Rathsfeld, and additionally established the invertibility of K + 1/2 on weighted L p -spaces for general polyhedra, see [15,30]. The Mellin techniques of [12,43] were adapted to the study of other layer potential operators in [32]. However, it appears that the reasoning in the proof of [32,Theorem 5] suffers from the same type of flaw as that of [43,Lemma 1.5], cf. Theorem 4.17 and Remark 4. 19. In [32] it was also proven that the essential spectral radius of K : L 2 (∂Ω) → L 2 (∂Ω) is less than 1/2 if Ω has sufficiently small Lipschitz character; the spectral radius conjecture asks if this is true for all Lipschitz domains. The spectral radius conjecture is known to be true on the Sobolev space H 1/2 (∂Ω) [9,49].

1.2.
Results. The main purpose of this article is to describe the essential spectrum of K : H 1/2 (∂Ω) → H 1/2 (∂Ω), or, equivalently, of K * : H −1/2 (∂Ω) → H −1/2 (∂Ω), for Lipschitz polyhedra Ω. Note that in the layer potential formulation (1.1) of the plasmonic problem, charges u ∈ H −1/2 (∂Ω) correspond to potentials U = Su with finite energy, R 3 |∇U | 2 dx < ∞. We will also investigate the spectra of K : Our results in this latter setting will serve as crucial tools for our study in H 1/2 (∂Ω), but we also believe that they are of some independent interest. All of our results in the L 2 α (∂Ω)-context are valid for arbitrary polyhedra, including non-Lipschitz polyhedra such as the interior of which is a variant of the so-called "two brick" domain. To understand general bounded polyhedra, we first analyze the Neumann-Poincaré operator K for polyhedral cones Γ which locally coincide with ∂Ω around vertices. Assuming that Γ has its vertex at the origin, we consider as in Figure 1 the spherical polygon where S 2 denotes the two-dimensional unit sphere. In this case, K is a Mellin operator with an operator-valued convolution kernel [42], and this leads us to consider the (direct value of the) double layer potential operator H(iξ) on γ, ξ ∈ R, formed with respect to the fundamental solution for −∆ S 2 + 1/4 + ξ 2 , where ∆ S 2 is the Laplace-Beltrami operator of S 2 -see Section 3.
In Theorem 5.27 and Lemma 4.13, we will describe the spectra of H(iξ) : , dω denoting the arc length measure on γ and q(ω) a quantity comparable to the distance from ω ∈ γ to the corners of γ. In particular, the essential spectrum in the former case is given by where β 1 , . . . , β J denotes the internal angles of γ, cf. (1.2). The remaining part of the spectrum σ H(iξ), H 1/2 (γ) consists of real eigenvalues. Let Similarly, for 0 ≤ α < 1 we introduce Λ α = {λ : λ is an isolated eigenvalue of H(z) : L 2 α (γ) → L 2 α (γ), for some Re z = 0}. It turns out that both of these sets are real, and that they can be equivalently formed by considering isolated eigenvalues of the L 2 (γ)-adjoint operators H * (z) : H −1/2 (γ) → H −1/2 (γ) and H * (z) : L 2 −α (γ) → L 2 −α (γ), respectively. It is an important observation that eigenfunctions to eigenvalues λ of H * (iξ) correspond to potentials U on S 2 such that and where λ and are related as in (1.1), see Lemma 3.10. Before we can discuss our main results, we need to introduce some additional notation. For Lipschitz polyhedral cones Γ, we introduce, following [39,Section 4], E(Γ) as a space of distributions on Γ with norm given by . By results of [39], E(Γ) is isomorphic to the L 2 (Γ)-dual of the homogeneous Sobolev spacė H 1/2 (Γ). We will see that K * : E(Γ) → E(Γ) is self-adjoint, and that E(Γ) is the correct space to consider for localization to H −1/2 (∂Ω). We letΓ denote the interior of Γ (which coincides locally with Ω around vertices), and we understand that [c, d) = (c, d] = ∅ if c > d. Let j = j * be the index which maximizes |1 − β j /π|.
One approach to the localization to ∂Ω is via the machinery of b-calculus [16,31,47], which relies on the construction of appropriate algebras of pseudo-differential operators. It may be possible to treat curvilinear polyhedra using such techniques. However, we shall take an alternative, rather direct approach to localization, staying within our scope of polyhedral domains (with flat faces). In one direction, we will construct Weyl sequences on ∂Ω in a procedure that seems applicable to a wider range of problems. We will prove complete localization results for both L 2 α (∂Ω), 0 ≤ α < 1, and H 1/2 (∂Ω), where L 2 α (∂Ω) is defined in analogy with L 2 α (Γ). We have chosen to state the result only in the case of H 1/2 (∂Ω) here, deferring the remaining statement to Theorem 4.21.
Theorem. Let K be the Neumann-Poincaré operator of a Lipschitz polyhedron ∂Ω. For each vertex of ∂Ω, let K i denote the Neumann-Poincaré operator of the corresponding tangent polyhedral cone Γ i , i = 1, . . . , I. Then, for λ ∈ C, K − λ is Fredholm on As in the definition of E(Γ), it is via the single layer potential possible to endow H −1/2 (∂Ω) with a scalar product that makes K * : H −1/2 (∂Ω) → H −1/2 (∂Ω) into a self-adjoint operator, see Section 2.3. Therefore the remaining non-essential spectrum of K : H 1/2 (∂Ω) → H 1/2 (∂Ω) consists of a sequence of isolated eigenvalues. Typically, eigenvalues can appear in the localization of operators, see [28] for a relevant illustration. However, we are not aware of any specific examples relevant to our setting.
Our description of σ ess (K, H 1/2 (∂Ω)) ∩ R + is particularly simple for convex polyhedra, since one only needs to consider the double layer potential of −∆ S 2 + 1/4 to compute this interval. Note that spectral parameters λ ∈ (0, 1/2) correspond to permittivities satisfying < −1; in [19,Section 8], it is suggested that < −1 is likely to be a necessary condition for the existence of surface plasmon waves on ∂Ω. Finally, we remark that the plasmonic problem for a cube is of importance to nanoplasmonics [14,17,20,25,45]. The numerics of [20] suggest that whenΓ is an octant of R 3 .

1.3.
Organization. In Section 2 we define the function spaces already mentioned, we recall some elements of Fredholm theory and extrapolation of compact operators, and we discuss Mellin operators with operator-valued convolution kernels. In Section 3 we examine the relationship between the Neumann-Poincaré operator on a polyhedral cone Γ and the double layer potential operators H(z) on the associated spherical polygon γ. Section 4 contains all of our theory concerning L 2 α (∂Ω), while Section 5 treats the energy space case.

Notation and Preliminaries
2.1. Polyhedral cones Γ. Throughout the paper, Ω will denote a simply connected bounded polyhedron with straight faces. The localization of K to a corner of ∂Ω leads us to consider integral operators on the boundary Γ of an infinite polyhedral coneΓ which locally coincides with Ω. Without loss of generality, we may assume that Γ has its vertex at the origin. The faces of Γ are open plane sectors F j , j = 1, . . . , J; the edges of Γ are denoted by υ j . We shall denote by γ the intersection of Γ with the two-dimensional unit sphere S 2 . That is, γ is a spherical polygon consisting of the circular arcs γ j = S 2 ∩ F j and the corner points E j = υ j ∩ S 2 , j = 1, . . . , J. Letγ = S 2 ∩Γ. Then ∂γ = γ andΓ = R +γ is the polyhedral cone with baseγ.
Let C ∞ arc (γ) be the set of all Lipschitz continuous functions on γ whose restrictions to γ j belong to C ∞ (γ j ), j = 1, . . . , J. In the vicinity of each corner E j , we parametrize the adjacent arcs γ j−1 and γ j by the arc lengths s = s j−1 and s = s j from E j . We fix a function q ∈ C ∞ arc (γ) such that q(ω) = s for ω = ω(s) in a neighborhood of each corner E j , non-vanishing except at corner points. For α ∈ R, we let where dω denotes the arc length measure on γ, and Note that the usual L 2 -space L 2 (Γ) coincides with L 2 (R + , r dr) ⊗ L 2 0 (γ).

2.2.
Weighted L 2 -spaces on ∂Ω. Throughout, ∂Ω will denote the boundary of the simply connected bounded polyhedron Ω ⊂ R 3 with vertices E i , i = 1, . . . , I and faces F j , j = 1, . . . , J. For each i = 1, . . . , I, let Γ i be the tangent polyhedral cone to ∂Ω at the corner E i . We define C ∞ face (∂Ω) as the space of Lipschitz continuous functions on ∂Ω that are C ∞ on the closure of each face F j . By a compactness argument, we can choose a partition of unity {ϕ i } I i=1 ⊂ C ∞ face (∂Ω) on ∂Ω, such that ϕ i ≡ 1 in a neighborhood of E i , ϕ i ≡ 0 in a neighborhood of j =i E j , and supp ϕ i ⊂ Γ i . Then, given a function f on ∂Ω, we can naturally understand ϕ i f as a function on Γ i . We define, for α < 1, the space L 2 α (∂Ω) as the completion of C ∞ face (∂Ω) in the norm f 2 2.3. The energy spaces on ∂Ω and Γ. Following an idea that dates back to Poincaré [9,23], the energy space E(∂Ω) is introduced as the Hilbert space obtained by completing L 2 (∂Ω) in the positive definite scalar product where S denotes the single layer potential on ∂Ω. The reason for introducing the energy space is that K * : E(∂Ω) → E(∂Ω) is self-adjoint, owing to the Plemelj formula SK * = KS. When discussing the energy space, we will for technical reasons assume that Ω is Lipschitz. That is, ∂Ω is locally the graph of a Lipschitz function whose epigraph locally coincides with Ω. Under this assumption, E(∂Ω) is a space of distributions on ∂Ω which is isomorphic to the Sobolev space H −1/2 (∂Ω) of index −1/2 along ∂Ω [49], with equivalent norms, . When Γ is a Lipschitz polyhedral cone, we similarly introduce the energy space E(Γ) as the completion of the space of compactly supported L 2 (Γ)-functions in the scalar product where S now denotes the single layer potential on Γ. In this case, E(Γ) coincides with the dual of the fractional homogeneous Sobolev spaceḢ 1/2 (Γ) on Γ [39, Theorem 14]. Now let Γ i , i = 1, . . . , I, be the tangent polyhedral cones to ∂Ω at the corners of ∂Ω, and let {ϕ i } I i=1 be the partition of unity on ∂Ω described in Section 2.2. Then, for f ∈ L 2 (∂Ω), supp ϕ i f ⊂ Γ i ∩ ∂Ω, and therefore . By density and the fact that each ϕ i is a multiplier of H −1/2 (∂Ω) E(∂Ω), it follows that 2.4. Fredholm theory. Let X and Y be Banach spaces and let T be a bounded linear operator from X to Y , that is, T ∈ L(X, Y ). Let α(T ) = dim Ker(T ) and β(T ) = dim(Y /Ran(T )), where Ker(T ) ⊆ X and Ran(T ) ⊆ Y denote the nullspace and the range of T , respectively. We say that T is a Fredholm operator if Ran(T ) is closed, α(T ) < ∞ and β(T ) < ∞. On the other hand, T is a upper semi-Fredholm operator if Ran(T ) is closed and α(T ) < ∞, whereas T is a lower semi-Fredholm operator if Ran(T ) is closed and β(T ) < ∞. If the operator T is either upper or lower semi-Fredholm, we shall say that it is semi-Fredholm. We will now recall some elements of Fredholm theory that will be useful for us. For a complete treatment, see [46]. The following criterion is very useful.
Then T is upper semi-Fredholm if and only if there is a Banach space Z, a compact operator S : X → Z, and a constant C > 0 such that A fundamental quantity associated with a (semi-)Fredholm operator is its index Proposition 2.2. Let X, Y be Banach spaces and T ∈ L(X, Y ) be a semi-Fredholm operator. If K is a compact operator from X to Y , then T +K is also semi-Fredholm and ind(T +K) = ind(T ).
Furthermore, the composition of Fredholm operators T and S is again Fredholm and ind(T S) = ind(T ) + ind(S). This formula is also true for semi-Fredholm operators, as long as the right-hand side makes sense.
For an operator T ∈ L(X) = L(X, X) we shall call Weyl sequence for an operator T ∈ L(X) and λ ∈ C, then λ ∈ σ ess (T, X). The converse is also true when X is a Hilbert space and T is self-adjoint.

Extrapolation of compactness.
When treating the energy space case in Section 5 we will sometimes rely on the extrapolation result of Cwikel, [10], in order to establish the compactness of certain operators. For a compatible couple of Hilbert spaces (A 0 , A 1 ) and 0 < θ < 1, let (A 0 , A 1 ) θ denote the real interpolation space between A 0 and A 1 . Since we are dealing with Hilbert spaces, we will always assume that q = 2 in the real interpolation method, omitting it from the notation. Also note that complex and real interpolation coincide in the Hilbert space case, see [8].
We will use extrapolation in the scale of Sobolev spaces, see [34].
Finally, we state the extrapolation result.

Mellin convolution operators.
Recall that the Mellin transform of a sufficiently nice function f : R + → C is defined by Up to a scaling factor, the Mellin transform induces a unitary map The Mellin convolution of appropriate functions f and g is given by and . Referring back to the notation for polyhedral cones introduced in Section 2.1, for a function f on Γ orΓ, we shall also write Mf (z) for the Mellin transform in the radial variable, Here x = rω has been written in spherical coordinates; r = dist(0, x) and ω ∈ γ or ω ∈γ.
We say that an operator T : When convergent, we shall then denote by MT (z) : C ∞ c (∪γ j ) → L 2 (γ) the operator given by the integral kernel We also make use of the analogous terminology and notation for operators T : Consider the multiplication operator M r 1/2 defined by Observe that if T is a Mellin convolution operator with kernel T (t, ω, ω ), then M r 1/2 T M r −1/2 is also a Mellin convolution operator with kernel t 1/2 T (t, ω, ω ). Therefore, at least formally, Furthermore, is unitary up to a scaling factor, where, for sufficiently nice f ∈ L 2 α (Γ) and Re z = 0, Let α < 1. Via (2.3), any Mellin convolution operator that extends to a bounded operator T : . With this in mind, we record the following elementary lemma for future use. We provide a proof to preserve the concrete presentation pursued in this section. Lemma 2.6. Let H be a separable Hilbert space and let {A(ξ)} ξ∈R be a strongly measurable family of operators A(ξ) : H → H such that sup ξ A(ξ) < ∞. Then defines a bounded operator, and Proof. Fix orthonormal bases {e j } j and {f k } k of L 2 (R, dξ) and H, respectively. Then every First assume that the sum is finite. Then The statement now follows in the usual manner, extending by density the domain of definition of I ⊗ A(ξ) from finite sums to arbitrary h.

Localizations of Mellin operators.
There is a well-developed theory also of pseudodifferential operators of Mellin type [11,27,33]. For our purposes, we only need to apply the results for a localized (scalar) Mellin convolution operator. The formulation described here can be deduced from Theorem 1 in [27] and the subsequent remark. Suppose that T is a Mellin convolution operator with a convolution kernel a(s/t) for which there are α < 0 < β such that sup α≤Re z≤β ) be a cut-off function such that ϕ ≡ 1 in a neighborhood of 0 and ϕ ≡ 0 in a neighborhood of 1. Then the essential spectrum of ϕT ϕ : Furthermore, when λ does not belong to this curve, then the Fredholm index of ϕT ϕ − λ coincides with the winding number of λ with respect to the essential spectrum. For −1 ≤ a < 1 and −3/2 < Re z < 3/2, we have that where the implied constant depends on z.
For −1 ≤ a < 1 and 1 < Re z < 2, we have that where, again, the implied constant depends on z.
3. The Mellin transform and layer potential operators on cones 3.1. Identification of MK(z). Let Γ be a polyhedral cone. In any study of the double layer potential on Γ, it is essential to analyze the Mellin transform MK(z), as seen in [12,42,43]. An explicit identification of MK(z) was made by Qiao and Nistor [42], in terms of layer potentials for Schrödinger operators on the spherical polygon γ. We will therefore recall a number of their calculations. We remind the reader that the single layer potential of −∆ on Γ is given by and that the direct value of the double layer potential of u on Γ is defined by where dS denotes the standard surface measure on Γ and n y is the outward normal vector at a.e. y ∈ Γ. In spherical coordinates, x = rω and y = r ω , where r = dist(0, x), r = dist(0, y), and ω, ω ∈ γ, we have that and Therefore S 0 := M r −1/2 SM r −1/2 is a Mellin convolution operator with operator-valued convolution kernel while K is itself a Mellin convolution operator with with kernel be the standard fundamental solution of −∆, understood as the operator defined by where x = rω, y = r ω , ω, ω ∈γ, and dS is the surface measure onγ.
Then Φ 0 is a Mellin convolution operator with kernel . On the other hand, the Laplacian in spherical coordinates is given by To understand the interaction between the Laplacian and the Mellin transform, note, for and that the Mellin transform and the Laplace-Beltrami operator commute, Noting that M r ∆M r = (r∂ r ) 2 + 3r∂ r + 2 + ∆γ, we find, applying (3.9) and (3.10),that This fundamental solution is unique, by the positive definiteness of −∆γ and the fact that We now recall one of the main results of [42]. See also [43,Lemma 3.2]. 42]). For −1/2 < Re z < 1/2, in terms of kernels, we have on γ that That is, MK(z + 1/2) is the direct value on γ of the double layer potential operator of −∆γ + 1/4 − z 2 .
3.2. The adjoint of MK(z) and the Kellogg argument on γ. Let H(z) = MK(z+1/2), −3/2 < Re z < 3/2, and let 0 ≤ α < 1. Recall from Section 2.6 that K : . For Re z = 0, we will in this section combine the description of H(z) as the double layer potential operator of −∆γ + 1/4 − z 2 with a variant of an argument due to O.D. Kellogg, in order to show that every eigenvalue of H * (z) : The success of the Kellogg argument relies on the fact that the potential 1/4 − z 2 > 0 for Re z = 0.
In this discussion, H * (z) = (H(z)) * denotes the adjoint of H(z) with respect to the L 2 (γ)pairing. For f ∈ L 2 α (γ) and g ∈ L 2 −α (γ), we see through the involution t → 1/t that where K † denotes the adjoint of K with respect to the L 2 0 (Γ)-pairing, which has operatorvalued convolution kernel Applying the argument of Section 2.6 again, we conclude the following.
. One could of course have arrived at this lemma directly by taking the adjoint in (3.11), but the preceding calculations are instructive for later arguments.
Therefore the equations remain valid for general g ∈ L 2 −α (γ), and in particular for g = f . Moreover, since f is an eigenfunction of H * (z), we have Remark 3.11. The proof in particular shows that for Re z = 0 and 0 = g ∈ L 2 (γ), This is the basis for constructing the energy space E(γ, −∆γ + 1/4 − z 2 ) H −1/2 (γ) in Section 5.1. Furthermore, once constructed, the proof of Lemma 3.10 shows that L 2 −α (γ) is continuously contained in E(γ, −∆γ +1/4−z 2 ), 0 ≤ α < 1. Dualizing, this can be interpreted as a non-sharp fractional Hardy inequality: 4. Spectral theory on L 2 α (∂Ω) 4.1. Analysis of H(z). Let Γ be a polyhedral cone and let K be its Neumann-Poincaré operator. Recall that we write H(z) = MK(z + 1/2), −3/2 < Re z < 3/2, so that Observe that H(z) is pointwise well-defined, since ω · n ω = 0 whenever ω and ω belong to the same arc γ j . For each corner E j of γ, we choose a function ϕ j ∈ C ∞ arc (γ) such that 0 ≤ ϕ j ≤ 1 on γ, ϕ j is supported in a small neighbourhood of E j , and ϕ j ≡ 1 close to E j . We then introduce the decomposition The starting point of this section is the following result of Elschner. We have extracted a slightly more precise statement than given in [12, Theorem 2.1], which follows from its proof. We denote the interior angle made by γ at E j by β j .
We shall also require the following lemma in our analysis.
Theorem 4.18 characterizes all real points in σ K, L 2 α (Γ) . There is a gap of complex points (4.20) because while H(z) − λ is invertible for every Re z = 0, we do not know how to control the resolvent (H(z) − λ) −1 uniformly in z, cf. Theorem 4.17.
It is thus sufficient to prove that the integral operator T with kernel T (rω, r ω ) is bounded on L 2 ([0, A] × γ, dr q(ω) −α dω). By applying the unitary transformation it is equivalent to prove that the integral operator T , To do this we verify that T is bounded on L 1 = L 1 ([0, A] × γ, r −1 dr q(ω) −1 dω) and L ∞ and apply the Riesz-Thorin interpolation theorem.
To see that it is bounded on L 1 , it is sufficient to see that By the change of variable h = r/r we have that Thus we are left to show that (4.22) sup Since 0 ≤ α < 1, we are only concerned with the situation that both ω and ω are close to the same corner of γ, and we then introduce arc-length parametrization. Thus we have to verify that sup 0<s <1 where we used that 0 ≤ α < 1. We conclude that (4.22) holds. The boundedness of T on L ∞ can be proved similarly.
Theorem 4.21. Let 0 ≤ α < 1 and let K be the Neumann-Poincaré operator of a polyhedron ∂Ω. For each vertex of ∂Ω, let K i denote the Neumann-Poincaré operator of the corresponding tangent polyhedral cone Γ i , i = 1, . . . , I.
The spectra σ(K i , L 2 α (Γ i )) have been described in Theorem 4.18. Remark 4.22. One would like to accompany this theorem with a Kellogg-type argument for K : L 2 α (∂Ω) → L 2 α (∂Ω), cf. Lemma 3.10, but we have been unable to produce such a result. Proof. Let {ϕ i } I i=1 be a partition of unity of ∂Ω, as described in Section 2.2, and let {η i } 1≤i≤I be a family of Lipschitz functions on ∂Ω such that η i ≡ 1 in a neighbourhood of supp ϕ i and η i ≡ 0 on ∂Ω \ Γ i . As usual, by very slight abuse of notation, we understand functions such as ϕ i and η i as functions on both ∂Ω and Γ i .
As a sequence (w n ) ⊂ L 2 α (∂Ω), w n L 2 (∂Ω) = 1 for every n and w n → 0 weakly. Furthermore, assuming we chose w n to have sufficiently small support, Here (1 − η i 0 )Kϕ i 0 w n → 0 in norm as n → ∞, since (1 − η i 0 )Kϕ i 0 is compact on L 2 α (∂Ω) and w n → 0 weakly, while because (w n ) is a Weyl sequence for (λI −K i 0 ). Thus (w n ) is a Weyl sequence also for λI −K, and therefore λI − K is not Fredholm. 5. Spectral theory on the energy space 5.1. The energy space of a polyhedral cone. Let Γ be a Lipschitz polyhedral cone. In this subsection we study the energy space E(Γ) defined in Section 2.3.
Lemma 5.23 implies that this identification depends continuously on ξ.
By the identification of E(Γ) in Section 5.1 and Lemma 5.25, we therefore obtain, for f, g ∈ C ∞ c (∪ j F j ), that In other words, we have the following lemma.
When the polyhedral cone is convex, we can obtain additional information about µ + .
We now provide our final theorem. The spectra σ(K * i , E(Γ i )) have been described in Theorem 5.29.
Proof. We follow the proof of Theorem 4.21, retaining its notation. Suppose that λI − K * i is invertible on E(Γ i ) for every i = 1, . . . , I. Then, for f ∈ E(∂Ω), where we have used that 1 − η i and ϕ i have disjoint support, and that By Lemmas 5.31 and 5.32, there is thus a Hilbert space H and a compact operator C : E(∂Ω) → H such that This shows that λI − K * is Fredholm on E(∂Ω), since K * : E(∂Ω) → E(∂Ω) is self-adjoint. Conversely, suppose that λI − K * i 0 : E(Γ i 0 ) → E(Γ i 0 ) fails to be invertible for some i 0 ∈ {1, . . . , I}. Then, by the proofs of Theorems 4.17 and 5.29, there is a singular Weyl sequence (w n ) for λI − K * i 0 : E(Γ i 0 ) → E(Γ i 0 ), supported in a sufficiently small neighborhood of the vertex of Γ i 0 . We interpret (w n ) as a sequence in E(∂Ω), tending to 0 weakly and satisfying that w n E(∂Ω) = 1 for all n, cf. Section 2.3.
Choosing the support of w n appropriately, we have the following equation in E(∂Ω), It is easy to verify that the operator (1 − η i 0 )K * ϕ i 0 : E(∂Ω) → E(∂Ω) is compact, either by extrapolation or by arguing as in the proof of Lemma 5.32. Therefore (1 − η i 0 )K * ϕ i 0 w n → 0 in E(∂Ω) as n → ∞, since w n → 0 weakly. Next we understand η i 0 (λI − K * )ϕ i 0 w n as an element of E(Γ i 0 ) satisfying Here (λI −K * i 0 )w n → 0 by the choice of (w n ) as a Weyl sequence, while (1−η i 0 )K * i 0 ϕ i 0 w n → 0 in E(Γ i 0 ) by Lemma 5.32.