A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields

TL;DR

This paper establishes a criterion linking the existence of zero modes of the Pauli operator in three dimensions to a specific quantity, for magnetic fields with particular decay, impacting inequalities in quantum mechanics.

Contribution

It introduces a new criterion based on a quantity δ(B) that determines zero mode existence for the Pauli operator with decaying magnetic fields.

Findings

Zero modes exist if and only if δ(B) = 0.

Sobolev, Hardy, and CLR inequalities hold without zero modes.

The result extends previous work to fields with decay rate |x|^{-2-β}.

Abstract

We consider the Pauli operator in $\mathbb R^3$ for magnetic fields in $L^{3/2}$ that decay at infinity as $|x|^{-2-\beta}$ with $\beta > 0$. In this case we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity $\delta(\mathbf B)$, defined below, being equal to zero. Complementing a result from [Balinsky, Evans, Lewis (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy and CLR inequalities hold whenever the magnetic field has no zero mode.