Energy Power Series Analysis of the Bound States of the One-dimensional Dirac Equation

Abstract

In this work we consider the one-dimensional Dirac equation including an electrostatic potential with compact support, and focus on the regime of bound states. We obtain exact expressions for both the characteristic function and the eigenfunctions in L 2 (ℝ, ℂ2), given in the form of power series of the energy parameter. This approach is meant for arbitrary bounded potentials, so that a square potential is a special case of the theory here presented. We derive an efficient numerical method for the calculation of approximate eigen-energies of the bound states. Finally, we investigate the physical sense of the eigen-energies that are forbidden in the non-relativistic regime in terms of the Klein tunneling.