  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.
