Pauli operator and Aharonov Casher theorem for measure valued magnetic fields

TL;DR

This paper extends the definition of the two-dimensional Pauli operator to measure-valued magnetic fields, bypassing traditional regularity conditions, and generalizes the Aharonov-Casher theorem to such singular fields.

Contribution

It introduces a new framework for defining the Pauli operator with measure-valued magnetic fields and extends the Aharonov-Casher theorem to these cases, including a counterexample for infinite total variation.

Findings

Extended the Pauli operator to measure-valued magnetic fields.

Generalized the Aharonov-Casher theorem for finite total variation measures.

Provided a counterexample for infinite total variation magnetic fields.

Abstract

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual $\bA\in L^2_{loc}$ condition on the vector potential which does not allow to consider such singular fields. We extend the Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted $L^2$ estimate on a singular integral operator.