A sharp criterion for zero modes of the Dirac equation

TL;DR

This paper establishes a precise mathematical criterion involving the $L^d$ norm of a vector potential for the existence of zero-energy solutions in the Dirac equation across different dimensions, and classifies all such solutions when the criterion is exactly met.

Contribution

It provides a sharp necessary condition for zero modes of the Dirac equation and classifies all vector potentials that admit zero modes at the critical norm.

Findings

Necessary condition involving $L^d$ norm for zero modes

Existence of zero modes at the critical norm in odd dimensions

Complete classification of vector potentials with zero modes

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.