Spectral structure of the Neumann–Poincaré operator on tori

Abstract

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey. Prerequisites of potential theoryWe recall in this section some terminology and basic facts of Newtonian potential theory. Details can be found in [6,56]. We warn the reader that there is no consensus in the vast literature on the subject of signs and constants in the definitions of potentials. We hope this will not be a cause of confusion.

Modal approximation for plasmonic resonators in the time domain: the scalar case

“…Using the Plemelj symmetrisation principle and the spectral theory of compact selfadjoint operators, the latter can be diagonalised in the appropriate functional spaces [24,35], which allows the scattered field to be decomposed in a basis of orthogonal modes in the static case [10]. The properties of the eigenvalues of the Neumann-Poincaré operator have been extensively studied in the literature, see the review paper [13] and references therein. For a smooth enough boundary, say C 1,α for some α > 0, the operator is compact and its eigenvalues are real numbers converging to zero.…”

Comparison of integral equations for the Maxwell transmission problem with general permittivities

“…Note that "Dirac" specified by (35), is better conditioned near x = −1, where false eigenwavenumbers occur both for "Dirac" and "HK 8-dens". For "Dirac" this false eigenwavenumber corresponds to the monopole field E = r/|r| 3 Our numerical results show that "Dirac" wins over "HK 8-dens" in almost all tests. "Dirac" does not have any false near-eigenwavenumbers for any passive materials, whereas "HK 8-dens" exhibits such in the plasmonic case, and even false eigenwavenumbers in the reverse plasmonic case.…”

“…of the acoustic double layer operator K ν k , appearing in the (1, 1) and (5, 5) diagonal blocks of E k . That is, K d equals the Neumann-Poincaré operator K NP , possibly modulo a sign depending on convention [3]. Its essential spectrum σ ess (K d ) in the fractional Sobolev space H 1/2 (Γ ), that is the set of λ for which λI − K d fails to be a Fredholm operator, is a compact subset of the interval (−1, 1), for any Lipschitz surface Γ .…”

Spectral properties of the Neumann-Poincaré operator on rotationally symmetric domains in two dimensions

“…Recently there is rapidly growing interest in the spectral properties of the NP operator in relation to the plasmonic resonance on meta material and significant new results are being produced. We refer to recent surveys [3,12] for historical account and recent development on the operator. In relation to the subject of this paper, we mention that K ∂Ω can be realized as a self-adjoint operator on H 1/2 (∂Ω) (H 1/2 (∂Ω) is the Sobolev space of order 1/2 on the curve ∂Ω) by introducing a new inner product [13].…”