On the spin-1/2 Aharonov-Bohm problem in conical space: bound states, scattering and helicity nonconservation

TL;DR

This paper investigates the quantum behavior of a spin-1/2 particle in a conical space with an Aharonov-Bohm potential, analyzing bound states, scattering, and helicity nonconservation using self-adjoint extension methods.

Contribution

It introduces a method to determine the self-adjoint extension parameter from physical parameters, providing explicit formulas for bound states and scattering in this context.

Findings

Derived explicit expressions for energy bound states and phase shifts.

Linked the self-adjoint extension parameter to physical parameters of the system.

Discussed the relation between bound states, zero modes, and helicity nonconservation.

Abstract

In this work the bound state and scattering problems for a spin-1/2 particle undergone to an Aharonov-Bohm potential in a conical space in the nonrelativistic limit are considered. The presence of a \delta-function singularity, which comes from the Zeeman spin interaction with the magnetic flux tube, is addressed by the self-adjoint extension method. One of the advantages of the present approach is the determination of the self-adjoint extension parameter in terms of physics of the problem. Expressions for the energy bound states, phase-shift and $S$ matrix are determined in terms of the self-adjoint extension parameter, which is explicitly determined in terms of the parameters of the problem. The relation between the bound state and zero modes and the failure of helicity conservation in the scattering problem and its relation with the gyromagnetic ratio $g$ are discussed. Also, as an application, we consider the spin-1/2 Aharonov-Bohm problem in conical space plus a two-dimensional isotropic harmonic oscillator.