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

TL;DR

The paper addresses the gap in Maxwell boundary-integral theory by introducing a Calderón-type symmetrization for the matrix Maxwell Neumann–Poincaré operator $\mathcal{M}_{\partial D}$ and its adjoint, establishing a self-adjoint framework on curl- and gradient-trace spaces and proving a spectral decomposition linked to the scalar NP operator. It shows that the spectra satisfy $\sigma(\mathcal{M}_{\partial D};\overset{\longrightarrow}{\mathrm{curl}}_{\partial D}(H^{\tfrac12}(\partial D)))=\sigma(\mathcal{K}^*_{\partial D})\setminus\{\tfrac12\}$ and $\sigma(\mathcal{M}^*_{\partial D};\nabla_{\partial D}(H^{\tfrac12}(\partial D)))=\sigma(-\mathcal{K}^*_{\partial D})\setminus\{-\tfrac12\}$, with eigenfunctions expressed via layer potentials. Using this spectral framework, the authors prove a quantum-ergodic boundary localization result for weak SPRs in the full Maxwell system and, in the spherical case, derive explicit exponential localization of plasmon modes. The work provides a rigorous operator-theoretic basis for predicting boundary-concentrated SPRs in arbitrary-shaped nanoparticles and yields concrete spherical-case formulas, advancing both the theory of boundary integral methods and applications in nanophotonics. These results have potential implications for plasmonic sensing, cloaking, and metamaterial design where precise control of surface localization is essential.

Abstract

The Neumann--Poincaré (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--Poincaré (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.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 2.1
• Lemma 2.2
• Theorem 2.3: Helmholtz Decomposition