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

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.