# Proper treatment of scalar and vector exponential potentials in the Klein-Gordon equation: Scattering and bound states

## TL;DR

This paper corrects previous misconceptions in solving the Klein-Gordon equation with exponential potentials, providing accurate solutions for bound and scattering states, and reveals a new charge density polarization effect.

## Contribution

It offers a corrected and generalized method for solving the Klein-Gordon equation with exponential potentials, including bound and scattering states, and introduces a novel polarization effect.

## Key findings

- Corrected solutions for bound states with exponential potentials
- Generalized approach for arbitrary scalar-vector coupling mixing
- Discovered a charge density polarization effect in weak potentials

## Abstract

We point out a misleading treatment in the literature regarding to bound-state solutions for the $s$-wave Klein-Gordon equation with exponential scalar and vector potentials. Following the appropriate procedure for an arbitrary mixing of scalar and vector couplings, we generalize earlier works and present the correct solution to bound states and additionally we address the issue of scattering states. Moreover, we present a new effect related to the polarization of the charge density in the presence of weak short-range exponential scalar and vector potentials.