On the Aharonov-Casher formula for different self-adjoint extensions of the Pauli operator with singular magnetic field

TL;DR

This paper investigates two self-adjoint extensions of the Pauli operator in a 2D quantum system with singular magnetic fields, proving an Aharonov-Casher formula for the maximal extension and demonstrating approximation by regular fields.

Contribution

It introduces and compares two self-adjoint extensions of the Pauli operator, establishing an Aharonov-Casher formula for the maximal extension and showing approximation by regular magnetic fields.

Findings

Aharonov-Casher formula proved for the maximal extension

Maximal extension can be approximated by operators with regular magnetic fields

Comparison of different self-adjoint extensions of the Pauli operator

Abstract

Two different self-adjoint Pauli extensions describing a spin-1/2 two-dimensional quantum system with singular magnetic field are studied. An Aharonov-Casher type formula is proved for the maximal Pauli extension and it is also checked that this extension can be approximated by operators corresponding to more regular magnetic fields.