Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class of Multiparameter Exponential-Type Potentials

Abstract

By using the exactly-solvable Schrödinger equation for a class of multi-parameter exponential-type potential, the analytical bound state solutions of the Klein-Gordon equation are presented. The proposal is based on the fact that the Klein-Gordon equation can be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal. The proposal has the advantage of avoiding the use of a specialized method to solve the Klein-Gordon equation for a specific exponential potential due that it can be derived by means of an appropriate choice of the involved parameters. For this, to show the usefulness of the method, the relativistic treatment of spinless particles subject to some already published exponential potentials are directly deduced and given as examples. So, beyond the particular cases considered in this work, this approach can be used to solve the Klein-Gordon equation for new exponential-type potentials having hypergeometric eigenfunctions. Also, it can be easily adapted to other approximations of the centrifugal term different to the Green-Aldrich used in this work.