Zero modes for the magnetic Pauli operator in even-dimensional Euclidean space

TL;DR

This paper investigates the existence of zero modes in the magnetic Pauli operator in even-dimensional Euclidean space, showing they do not exist if the magnetic field decays faster than quadratically, but may appear if it decays quadratically.

Contribution

It provides new conditions for the existence of zero modes in the magnetic Pauli operator, correcting previous misconceptions from earlier literature.

Findings

No zero modes if magnetic field decays faster than quadratically.

Zero modes may appear if magnetic field decays quadratically.

Provides a lower bound for the number of zero modes in the quadratic decay case.

Abstract

We study the ground state of the Pauli Hamiltonian with a magnetic field in R^(2d). We consider the case where a scalar potential W is present and the magnetic field B is given by $B=2i\partial\bar\partial W$. The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.