Faraday effect revisited: sum rules and convergence issues

TL;DR

This paper revisits the Faraday effect, focusing on the convergence of sum rules related to the Verdet constant in solid state physics, and discusses mathematical issues arising from regularity assumptions.

Contribution

It provides a comprehensive analysis of convergence issues in sum rules for the Faraday effect, especially for smooth potentials, and highlights open problems for less regular cases.

Findings

Sum rules are crucial for understanding the Faraday effect.

Convergence of sums over band indices depends on potential regularity.

Open problems remain for less regular perturbations.

Abstract

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 an open problem related to less regular perturbations.