  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$.
