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

Abstract

We consider the Neumann–Poincaré 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–Poincaré operator. On both the crescent-shaped domain and touching disks, the Neumann–Poincaré 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.