Some paradoxes associated with a recent sum rule in scattering theory

This paper reports on some peculiarities associated with a recently published summation rule for scattering of electromagnetic
waves. The summation rule states that the extinction cross section integrated over all frequencies is equal to
the low-frequency response of the target. Although the summation rule is intriguing by itself, it becomes even more
paradoxical when a static conductivity model or the PEC boundary condition is introduced. The paradoxical character
lies in the fact that the extinction cross section integrated over all frequencies is independent of the static conductivity
as long as it is non-zero. This puzzling result is explained by rejecting the static conductivity model at zero frequency
as suggested by numerical simulations of a homogeneous and isotropic sphere. In addition, the low-frequency behavior
of diamagnetic materials is investigated using Herglotz functions and arguments from the theory of special relativity.