Paradox of multiple plasmonic resonances at light scattering by a cylinder of infinitesimal radius

The paradox of the divergence of the resonant scattering cross section of a cylinder with the permittivity equals minus unity and vanishing radius (R) irradiated by a monochromatic electromagnetic wave is discussed. Within the framework of the exact solution of the Maxwell equations, the divergence at the specified conditions is caused by the overlap of all but one multipolar resonances. It is shown that the paradox is caused by the too straightforward analysis of the expression for the cross section, which has a singularity at this point. To resolve the singularity, one must, first, generalize the problem formulation, taking into account the final linewidth of the incident wave, and then perform the correct sequence of limit transitions. The application of this approach gives rise to the vanishing cross section at the vanishing R. It ruins the expectations to employ such a cylinder as a superscatterer but simultaneously open a door to counterintuitive effects both in far and near field zones related to unusual size dependences of the scattered fields at small but finite R.