A sharp criterion for zero modes of the Dirac equation

Abstract

It is shown that $\Vert A{\Vert }_{L^d}^2\ge \frac{d}{d-2}{S}_d$ is a necessary condition for the existence of a nontrivial solution $\psi$ of the Dirac equation $\gamma \cdot (-i\nabla -A)\psi =0$ in $d$ dimensions. Here, $S_d$ is the sharp Sobolev constant. If $d$ is odd and $\Vert A{\Vert }_{L^d}^2=\frac{d}{d-2}{S}_d$, then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.