Faraday effect revisited: sum rules and convergence issues

This is the third paper of a series revisiting the Faraday effect. The question of the absolute convergence of the sums over the band indices entering the Verdet constant is considered. In general, sum rules and traces per unit volume play an important role in solid-state physics, and they give rise to certain convergence problems widely ignored by physicists. We give a complete answer in the case of smooth potentials and formulate a number of open problems related to less regular perturbations.