Bound states of a Klein-Gordon particle in presence of a smooth potential well

TL;DR

This paper analytically solves the one-dimensional Klein-Gordon equation with a smooth potential well, deriving bound state solutions using Whittaker functions and discussing antiparticle states based on potential parameters.

Contribution

It provides an exact analytical solution for bound states of a Klein-Gordon particle in a smooth potential well, including antiparticle state analysis.

Findings

Bound states expressed via Whittaker functions

Antiparticle bound states depend on potential parameters

Analytical solutions for Klein-Gordon in smooth wells

Abstract

We solve the one-dimensional time-independent Klein-Gordon equation in presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker $M_{\kappa,\mu}(x)$ function, and the antiparticle bound state is discussed in terms of potential parameters.