Quantum integrable systems and concentration of plasmon resonance

TL;DR

This paper provides a rigorous mathematical analysis of surface plasmon resonance (SPR) concentration at high-curvature interfaces, using quantum integrable systems and spectral theory to derive explicit characterizations and ergodicity results.

Contribution

It introduces a novel framework combining quantum ergodicity and integrable systems to analyze SPR concentration, extending spectral theory methods.

Findings

Sharper characterizations of SPR concentration at high-curvature points.

Establishment of quantum ergodicity results for the NP operator system.

Extension of quantum integrable system theory with potential broader applications.

Abstract

We are concerned with the quantitative mathematical understanding of surface plasmon resonance (SPR) when $d \geq 3$. SPR is the resonant oscillation of conducting electrons at the interface between negative and positive permittivity materials and forms the fundamental basis of many cutting-edge applications of metamaterials. It is recently found that the SPR concentrates due to curvature effect. In this paper, we derive sharper and more explicit characterisations of the SPR concentration at high-curvature places in both the static and quasi-static regimes. The study can be boiled down to analyzing the geometries of the so-called Neumann-Poincar\'e (NP) operators, which are certain pseudo-differential operators sitting on the interfacial boundary. We propose to study the joint Hamiltonian flow of an integral system given by the moment map defined by the NP operator. Via considering the Heisenberg picture and lifting the joint flow to a joint wave propagator, we establish a more general version of quantum ergodicity on each leaf of the foliation of this integrable system, which can then be used to establish the desired SPR concentration results. The mathematical framework developed in this paper leverages the Heisenberg picture of quantization and extends some results of quantum integrable system via generalising the concept of quantum ergodicity, which can be of independent interest to the spectral theory and the potential theory.