A sharp criterion for zero modes of the Dirac equation

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 of the Dirac equation $γ\cdot (-i\nabla -A)ψ= 0$ 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.

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