A sharp criterion for zero modes of the Dirac equation

Rupert L. Frank; Michael Loss

doi:10.4171/jems/1475

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A sharp criterion for zero modes of the Dirac equation

Rupert L. Frank
Ludwig-Maximilians Universität München, München, Germany; Munich Center for Quantum Science and Technology, München, Germany; California Institute of Technology, Pasadena, USA
Michael Loss
Georgia Institute of Technology, Atlanta, USA

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Abstract

It is shown that $∥ A ∥_{L^{d}}^{2} \geq \frac{d}{d - 2} S_{d}$ is a necessary condition for the existence of a nontrivial solution $ψ$ 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 $∥ A ∥_{L^{d}}^{2} = \frac{d}{d - 2} S_{d}$ , then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

Cite this article

Rupert L. Frank, Michael Loss, A sharp criterion for zero modes of the Dirac equation. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1475