Self-adjoint Extension Approach to the spin-1/2 Aharonov-Bohm-Coulomb Problem

TL;DR

This paper investigates the spin-1/2 Aharonov-Bohm-Coulomb problem using self-adjoint extension methods, revealing conditions for singular solutions and deriving bound state energies in a quantum system with Coulomb and magnetic flux effects.

Contribution

It introduces a self-adjoint extension approach to analyze the spin-1/2 Aharonov-Bohm-Coulomb problem, identifying when singular solutions occur and providing explicit energy expressions.

Findings

Singular solutions occur only for half the flux parameter range.

Bound state energies are explicitly derived.

Conditions for singular solutions with generalized potentials are established.

Abstract

The spin-1/2 Aharonov-Bohm problem is examined in the Galilean limit for the case in which a Coulomb potential is included. It is found that the application of the self-adjoint extension method to this system yields singular solutions only for one-half the full range of flux parameter which is allowed in the limit of vanishing Coulomb potential. Thus one has a remarkable example of a case in which the condition of normalizability is necessary but not sufficient for the occurrence of singular solutions. Expressions for the bound state energies are derived. Also the conditions for the occurrence of singular solutions are obtained when the non-gauge potential is $\xi/r^p (0\leq p<2)$.