Weakly (and not so weakly) bound states of a relativistic particle in one dimension

TL;DR

This paper provides an exact perturbative calculation of bound state energies for a relativistic one-dimensional particle in weak potentials, demonstrating the effectiveness of Padé approximants for extending results to stronger potentials.

Contribution

It presents the first exact fourth-order perturbation theory for relativistic bound states in one dimension and extends the analysis to two dimensions with confinement.

Findings

Perturbation theory accurately predicts bound state energies in weak potentials.

Padé approximants improve estimates for deep potential wells.

Analytical results agree well with numerical simulations for Gaussian wells.

Abstract

We present the first exact calculation of the energy of the bound state of a one dimensional Dirac massive particle in weak short-range arbitrary potentials, using perturbation theory to fourth order (the analogous result for two dimensional systems with confinement along one direction and arbitrary mass is also calculated to second order). We show that the non--perturbative extension obtained using Pad\'e approximants can provide remarkably good approximations even for deep wells, in certain range of physical parameters. As an example, we discuss the case of two gaussian wells, comparing numerical and analytical results, predicted by our formulas.