Spectral analysis of the Neumann-Poincare operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance

Abstract

We consider the Neumann-Poincare operator on a planar domain enclosed by two touching circular boundaries. This domain, which is a crescent-shaped domain or touching disks, has a cusp at the touching point of two circles. We analyze the operator via the Fourier transform on the boundary circles of the domain. In particular, we define a Hilbert space on which the operator is bounded, self-adjoint. We then obtain the complete spectral resolution of the Neumann- Poincare operator. On both the crescent-shaped domain and touching disks, the Neumann-Poincare operator has only absolutely continuous spectrum on the closed interval [-1/2, 1/2]. As an application, we analyze the plasmon resonance on the crescent-shaped domain.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).