Quantum ergodicity and localization of plasmon resonances. (English) Zbl 1534.82029
Summary: We are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a non-radiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is very sensitive to the material interface. However, we show that the SPR oscillation asymptotically localizes at places with high magnitude of curvature in a certain sense under an assumption equivalent to convexity in the three-dimensional setting. Our work leverages the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdière and Helffer-Martinez-Robert, as well as certain novel and more general ergodic properties of the Neumann-Poincaré operator to analyze the SPR field, which are of independent interest to the spectral theory and the potential theory.
MSC:
| 82D80 | Statistical mechanics of nanostructures and nanoparticles |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
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