# Bound state solutions of the Klein Gordon equation with energy-dependent potentials

## TL;DR

This paper derives exact bound state solutions for the Klein-Gordon equation with an energy-dependent Coulomb-like potential, introducing a novel energy-dependent mass function and analyzing various potential configurations.

## Contribution

It presents a new approach by considering an energy-dependent mass function and provides analytic solutions for different potential cases using the asymptotic iteration method.

## Key findings

- Derived explicit energy spectra for energy-dependent potentials
- Identified relations among tuning parameters affecting solutions
- Validated wave functions satisfy boundary conditions

## Abstract

In this manuscript, we investigate the exact bound state solution of the Klein-Gordon equation for an energy-dependent Coulomb-like vector plus scalar potential energies. To the best of our knowledge, this problem is examined in literature with a constant and position dependent mass functions. As a novelty, we assume a mass-function that depends on energy and position and revisit the problem with the following cases: First, we examine the case where the mixed vector and scalar potential energy possess equal magnitude and equal sign as well as an opposite sign. Then, we study pure scalar and pure vector cases. In each case, we derive an analytic expression of the energy spectrum by employing the asymptotic iteration method. We obtain a non-trivial relation among the tuning parameters which lead the examined problem to a constant mass one. Finally, we calculate the energy spectrum by the Secant method and show that the corresponding unnormalized wave functions satisfy the boundary conditions. We conclude the manuscript with a comparison of the calculated energy spectra versus tuning parameters.