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.