Resonance frequencies of arbitrarily shaped dielectric cylinders

We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence of resonance (scattering) frequencies associated with these eigenvalues is proved. The technique involves the determination of characteristic numbers (CNs) of the Fredholm operator-valued functions of the problems constructed using Green's potentials. Separating principal parts in the form of meromorphic operator pencils, we apply the operator generalization of Rouche's theorem to verify the occurrence of CNs in close proximities of the pencil poles. The results are illustrated in detail using the case of a dielectric cylinder of circular cross section.