Bound states by a pseudoscalar Coulomb potential in 1+1 dimensions

TL;DR

This paper solves the Dirac equation with a pseudoscalar Coulomb potential in 1+1 dimensions, revealing bound states absent in 3+1 dimensions, and explores general power-law potentials and their bound solutions.

Contribution

It provides the first analysis of bound states for pseudoscalar Coulomb potentials in 1+1 dimensions and examines the effects of various power-law potentials on bound solutions.

Findings

Bound states exist in 1+1 dimensions for pseudoscalar Coulomb potentials.

No bound states are found in 3+1 dimensions for the same potential.

Nonsingular power-law potentials can lead to bounded solutions.

Abstract

The Dirac equation is solved for a pseudoscalar Coulomb potential in a two-dimensional world. An infinite sequence of bounded solutions are obtained. These results are in sharp contrast with those ones obtained in 3+1 dimensions where no bound-state solutions are found. Next the general two-dimensional problem for pseudoscalar power-law potentials is addressed consenting us to conclude that a nonsingular potential leads to bounded solutions. The behaviour of the upper and lower components of the Dirac spinor for a confining linear potential nonconserving- as well as conserving-parity, even if the potential is unbounded from below, is discussed in some detail.