Bounded solutions of fermions in the background of mixed vector-scalar P\"{o}schl-Teller-like potentials

TL;DR

This paper investigates bound-state solutions of fermions in two-dimensional space with mixed vector-scalar Pöschl-Teller-like potentials, revealing unusual behaviors and connections to massless fermions in topological backgrounds.

Contribution

It provides an exact solution for fermions in a specific mixed potential, mapping the Dirac equation to a solvable Sturm-Liouville problem and exploring the delta potential limit.

Findings

Bound states are explicitly obtained for the modified Pöschl-Teller potential.

Unusual behaviors of Dirac spinor components are analyzed.

The problem relates to massless fermions in topological scalar and pseudoscalar fields.

Abstract

The problem of a fermion subject to a convenient mixing of vector and scalar potentials in a two-dimensional space-time is mapped into a Sturm-Liouville problem. For a specific case which gives rise to an exactly solvable effective modified P\"{o}schl-Teller potential in the Sturm-Liouville problem, bound-state solutions are found. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed. The Dirac delta potential as a limit of the modified P% \"{o}schl-Teller potential is also discussed. The problem is also shown to be mapped into that of massless fermions subject to classical topological scalar and pseudoscalar potentials.