Scattering solutions of the Klein Gordon equation for a step potential with hyperbolic tangent potential

TL;DR

This paper derives scattering solutions for the Klein-Gordon equation with a step and hyperbolic tangent potential, revealing phenomena like superradiance and transmission resonances through hypergeometric functions.

Contribution

It provides exact analytical solutions for the Klein-Gordon equation in this potential, including calculations of reflection and transmission coefficients.

Findings

Observation of superradiance

Identification of transmission resonances

Analytical solutions using hypergeometric functions

Abstract

We solve the Klein-Gordon equation for a step potential with hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection coefficient R and transmission coefficient T are calculated, we observed superradiance and transmission resonances.