On the non-existence of zero modes

TL;DR

This paper proves that certain magnetic fields aligned with a conformal Killing field, specifically those generating a simple rotation, cannot produce zero modes in Weyl-Dirac operators, expanding known classes of non-zero mode fields.

Contribution

It establishes a non-existence result for zero modes in Weyl-Dirac operators under specific magnetic field configurations related to conformal Killing fields.

Findings

Magnetic fields parallel to a simple rotation conformal Killing field do not produce zero modes.

The class of magnetic fields known not to generate zero modes is expanded to include smooth, compactly supported examples.

The result applies to a broad class of magnetic fields aligned with conformal symmetries.

Abstract

We consider magnetic fields on $\mathbb{R}^3$ which are parallel to a conformal Killing field. When the latter generates a simple rotation we show that a Weyl-Dirac operator with such a magnetic field cannot have a zero mode. In particular this allows us to expand the class of non zero mode producing magnetic fields to include examples of non-trivial smooth compactly supported fields.