Bounded solutions of fermions in the background of mixed vector-scalar inversely linear potentials

TL;DR

This paper finds exact bounded solutions for fermions in a two-dimensional setting with mixed vector-scalar inversely linear potentials, revealing unusual behaviors and implications for nonrelativistic hydrogen-like systems.

Contribution

It provides a closed-form solution for fermions in a mixed potential, extending previous scalar-only analyses and exploring the nonrelativistic limit implications.

Findings

Exact solutions for fermions in inversely linear potentials

Unusual behaviors of Dirac spinor components

Support for nonexistence of even-parity solutions in nonrelativistic hydrogen

Abstract

The problem of a fermion subject to a general mixing of vector and scalar potentials in a two-dimensional world is mapped into a Sturm-Liouville problem. Isolated bounded solutions are also searched. For the specific case of an inversely linear potential, which gives rise to an effective Kratzer potential in the Sturm-Liouville problem, exact bounded solutions are found in closed form. The case of a pure scalar potential with their isolated zero-energy solutions, already analyzed in a previous work, is obtained as a particular case. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed. The nonrelativistic limit of our results adds a new support to the conclusion that even-parity solutions to the nonrelativistic one-dimensional hydrogen atom do not exist.