Symmetrization of the Maxwell--Neumann--Poincar'e operator, spectral decomposition in $\mathbf{H}(\mathrm{curl},D)$ traces, and boundary localisation of SPRs

TL;DR

This paper introduces a symmetrization and spectral decomposition for the Maxwell Neumann--Poincaré operator, advancing understanding of surface plasmon resonances in electromagnetic systems and their boundary localization.

Contribution

It develops a novel symmetrization principle for the matrix-valued Maxwell NP operator and characterizes boundary localization of SPRs in the full Maxwell system.

Findings

Spectral decomposition of the Maxwell NP operator in $ extbf{H}( ext{curl},D)$ traces.

Rigorous characterization of boundary localization of SPRs.

Resolution of a long-standing question on quantitative SPR description.

Abstract

The Neumann--Poincar\'{e} (NP) operator, a fundamental operator in potential theory, has attracted renewed attention for its central role in the analysis of surface plasmon resonances (SPRs). SPRs, characterized by non-radiative electromagnetic waves at material interfaces with opposing permittivities, underpin advanced technologies such as bio-sensing and cloaking devices. While spectral properties of the scalar NP operator and SPR dynamics for scalar waves are well-established, their vectorial counterparts in Maxwell's framework remain poorly understood. This work bridges this gap by introducing a novel symmetrization principle for the matrix-valued Maxwell Neumann--Poincar\'{e} (MNP) operator, enabling a spectral decomposition of traces in the $\mathbf{H}(\mathrm{curl},D)$ space--a foundational advance for electromagnetic theory. Building on this framework, we rigorously characterize the quantum-ergodic localization of weak surface plasmon resonances at material boundaries in the full Maxwell system, thereby settling a long-standing question concerning their quantitative description.