Scattering of a scalar relativistic particle by the hyperbolic tangent potential

TL;DR

This paper analytically solves the Klein-Gordon equation with a hyperbolic tangent potential, deriving scattering solutions and calculating reflection and transmission coefficients, including conditions for superradiance.

Contribution

It provides exact analytical solutions for relativistic scattering in a hyperbolic tangent potential, extending understanding of superradiance phenomena in such systems.

Findings

Derived explicit scattering solutions using hypergeometric functions

Calculated reflection and transmission coefficients in closed form

Identified conditions under which superradiance occurs

Abstract

We solve the Klein-Gordon equation in the presence of the hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection $R$ and transmission $T$ coefficients are calculated in terms of Gamma function and, superradiance is discussed, when the reflection coefficient $R$ is greater than one.