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 $\sum_λ|λ|^2|1+2λ|^4$ is convergent, counting multiplicities of the eigenvalues $λ$. This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity $ε_r=-1$.

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