The quasi-static plasmonic problem for polyhedra

TL;DR

This paper characterizes the essential spectrum of the plasmonic problem for polyhedra in three-dimensional space, focusing on convex shapes and specific permittivity values, using boundary integral operators.

Contribution

It provides a spectral characterization of the plasmonic problem for polyhedra, especially for convex cases and certain permittivities, via analysis of the Neumann--Poincaré operator.

Findings

Simplified spectral description for convex polyhedra

Explicit characterization for permittivity $ extless -1$

Analysis of the double layer potential for polyhedral cones

Abstract

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.