Spectral theory of plasmonic resonances in the nonlocal regime

TL;DR

This paper provides a rigorous spectral analysis of nonlocal plasmonic resonances, revealing a fundamental change in spectral structure that regularizes singularities and alters the number of surface plasmon modes.

Contribution

It introduces a boundary integral framework for analyzing nonlocal plasmonic eigenvalues, showing a discrete real spectrum with finitely many modes, unlike the local theory.

Findings

Nonlocal model has a discrete real spectrum with a single accumulation point.

Only finitely many surface plasmon modes exist in the nonlocal regime.

Nonlocality regularizes field singularities in sharp geometries.

Abstract

We present a rigorous spectral analysis of plasmonic resonances in the nonlocal regime of spatially dispersive media. We adopt the quasi-static approximation of the hydrodynamic Drude model, which provides an analytically tractable setting to account for nonlocal effects. By reformulating the governing equations as a boundary integral system, we obtain an analytic Fredholm operator pencil that characterizes resonant behaviour. This framework enables the study of nonlocal plasmonic eigenvalues in general bounded Lipschitz domains, together with a corresponding resonant expansion of the scattered field. Our main result reveals a fundamental change in spectral structure: in contrast to the local theory -- which exhibits infinitely many surface plasmon modes and field singularities in domains with corners -- the nonlocal model admits a discrete real spectrum with a single accumulation point at a strictly positive value. Consequently, only finitely many surface plasmon modes exist, and the singular behaviour associated with sharp geometries is regularized. As such, our results show that nonlocality does not merely shift plasmonic resonances but completely reshapes the spectral landscape in a way that provides a mathematically transparent explanation for several nonlocal phenomena previously observed in numerical studies.