Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics

Abstract

Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling the interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact and Hilbert--Schmidt operator polynomials are constructed from the Green operator of electromagnetic scattering on scatterers with smooth boundaries. As a consequence, it is shown that the strongly singular Born equation has a discrete spectrum, and that the spectral series ${\sum }_{\lambda }|\lambda {|}^2|1+2\lambda {|}^4$ is convergent, counting multiplicities of the eigenvalues $\lambda$. This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity ${\epsilon }_r=-1$.

(v1) 86 pages, 2 figures; (v2) 57 pages, 3 figures. Title changed. Abridged and updated from arXiv:0911.4540v1, incorporating a considerable amount of materials from a subset of arXiv:1007.4375v2; (v3,v4) 31 pages. Abridged from arXiv:0911.4540v2 and arXiv:1007.4375v2. (v5) 32 pages. Revised according to referees' reports

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