A sharp criterion for zero modes of the Dirac equation
Abstract
It is shown that $\Vert A \Vert_{L^d}^2 \ge \frac{d}{d2}\, S_d$ 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 $\Vert A \Vert_{L^d}^2= \frac{d}{d2}\, S_d$, then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.03610
 arXiv:
 arXiv:2201.03610
 Bibcode:
 2022arXiv220103610F
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Primary: 35F50;
 Secondary: 81V45;
 47J10
 EPrint:
 LaTeX, 26 pages