Remarks on the Dirac Kepler/Coulomb problem with a space-dependent mass term

We investigate the Dirac Kepler/Coulomb (KC) problem in (2 + 1)-dimensions where the charged particle (an electron) is simultaneously affected by a position-dependent effective mass term. In polar coordinates, we examine two instances: (i) the quantum dynamics of an electron moving in the presence of both a Coulomb potential and a Coulomb-like scalar potential. This problem turns to be exactly solvable; (ii) the planar dynamics of a self-interacting electron in a Coulomb potential where nonlinearities are introduced via a scalar self-interaction, known as the Soler model. We obtain the approximate (nonlinear) ground-state energy levels together with their respective eigenfunctions. Thus, within the same framework, we present two different relativistic quantum-mechanical models which can be further explored.

Automated summary

In this research, the Dirac Kepler/Coulomb (KC) problem in (2 + 1)-dimensions with a position-dependent effective mass term was investigated. By examining two instances in polar coordinates, two different relativistic quantum-mechanical models were presented, involving the quantum dynamics of an electron in the presence of Coulomb potential and a Coulomb-like scalar potential, as well as the planar dynamics of a self-interacting electron in a Coulomb potential with nonlinearities introduced via a scalar self-interaction.