Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics

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 ${\sum }_{\lambda }\mathrm{\mid }\lambda {\mathrm{\mid }}^2\mathrm{\mid }1+2\lambda {\mathrm{\mid }}^4$ is convergent, counting multiplicities of the eigenvalues $\lambda$. This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity ${ϵ}_r=-1$.