Which magnetic fields support a zero mode?

TL;DR

This paper investigates the conditions under which magnetic fields support zero modes in the three-dimensional Dirac equation, establishing a norm-based criterion that highlights the importance of the spinorial nature of wave functions.

Contribution

It provides a new bound relating the magnetic field's norm to the existence of zero modes, emphasizing the role of spinorial properties and deriving an improved diamagnetic inequality.

Findings

Magnetic fields with a 3/2-norm greater than twice the Sobolev constant support zero modes.

The spinorial nature of wave functions is crucial for the derived bounds.

Results are not necessarily sharp, but some analyzed equations achieve optimal bounds.

Abstract

This paper presents some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. It is a well known fact that for the Schr\"odinger in three dimensions to have a negative energy bound state, the 3/2- norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3/2 - norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.