On the Pauli operator for the Aharonov-Bohm effect with two solenoids

TL;DR

This paper analyzes the Pauli operator for a spin-1/2 particle influenced by two Aharonov-Bohm fluxes, establishing boundary conditions, constructing a basis in deficiency subspaces, and enabling the application of Krein's formula.

Contribution

It explicitly constructs a basis in deficiency subspaces for the Pauli operator with two fluxes, facilitating spectral analysis via Krein's formula.

Findings

Defined boundary conditions at the vortices for the Pauli operator.

Constructed a basis in deficiency subspaces for the symmetric operator.

Enabled the use of Krein's formula for spectral analysis.

Abstract

We consider a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov-Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein's formula to the Pauli operator.