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arXiv · 2201.03610

A sharp criterion for zero modes of the Dirac equation

Rupert L. Frank·Michael Loss
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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 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.

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Rupert L. Frank, Michael Loss. 2022-01-10. A sharp criterion for zero modes of the Dirac equation. https://arxiv.org/abs/2201.03610

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