Exact solution for a fermion in the background of a scalar inversely linear potential

TL;DR

This paper derives exact solutions for a fermion in a two-dimensional scalar inverse linear potential, revealing novel behaviors of Dirac spinor components and expanding understanding of such quantum systems.

Contribution

It provides the first exact closed-form solutions for a fermion in a scalar inverse linear potential, linking it to the Kratzer potential and analyzing Dirac spinor components.

Findings

Exact bounded solutions for the inverse linear potential case.

Unusual behaviors of Dirac spinor components observed.

Connection established between inverse linear and Kratzer potentials.

Abstract

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials. The normalizable zero-eigenmode solutions are also searched. For the specific case of an inversely linear potential, which gives rise to an effective Kratzer potential, exact bounded solutions are found in closed form. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed.