The exterior Calderón operator for non-spherical objects

Abstract

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. IntroductionThe 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 ().,-volV) (0123456789().,-volV) 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 scattere...

On a Steklov spectrum in Electromagnetics

“…Conversely, by the Fundamental Lemma of the Calculus of Variations (see e.g., [3]), one can see that if E is a smooth solution of (26) then it is also a solution of (25) in the classical sense.…”

“…where k 2 := ω 2 εµ, and as usual we assume that Imk ≥ 0. Instead of the standard interior Calderón operator (see e.g., [8], [26]), in what follows we consider its variant defined by m → (ν × H) × ν, i.e.…”

On an interior Calderón operator and a related Steklov eigenproblem for Maxwell's equations

“…Another approach to the representation of an exterior Calderón operator associated with a scattering problem for not necessarily spherical domains is proposed in [19]; there the appropriate series expansions are performed with respect to generalized harmonics (the set of eigenfunctions to the Laplace-Beltrami operator for the domain's boundary). Further, the norm in an appropriate trace space of the exterior Calderón operator is obtained in view of an eigenproblem for a suitable quadratic form.…”

On an Interior Calderón Operator and a Related Steklov Eigenproblem for Maxwell's Equations

“…Another approach to the representation of an exterior Calder\' on operator associated with a scattering problem for not necessarily spherical domains is proposed in [19]; there the appropriate series expansions are performed with respect to generalized harmonics (the set of eigenfunctions to the Laplace--Beltrami operator for the Downloaded 11/21/20 to 147.162.22.66. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms domain's boundary).…”