Nonlinear Klein-Gordon equation and the Bose-Einstein condensation

TL;DR

This paper investigates solutions to the nonlinear Klein-Gordon equation with Coulombic and harmonic potentials, exploring relativistic effects and self-interactions in bosonic systems relevant to cosmology and high-energy physics.

Contribution

It provides a detailed analysis of Klein-Gordon solutions under different potentials using the Feshbach-Villars method, including effects of self-interaction and relativistic limits.

Findings

Solutions converge to Schrödinger equation results as relativistic effects diminish.

Self-interacting particles influence the harmonic potential solutions.

Relativistic effects are significant in certain potential regimes.

Abstract

The interest in the Klein-Gordon equation with different potentials has increased in recent years due to its possible applications in Cosmology, Hadron Physics and High-Energy Physics. In this work we investigate the solutions of the Klein-Gordon equation for bosons under the influence of an external potential by using the Feshbach-Villars method. We present detailed results for two cases: the Coulombic potential and the harmonic potential. For the latter case, we studied the effects of self-interacting particles by adopting a mean-field approach. We show that our results converge smoothly to the solution of the Schr\"odinger equation for the same systems as the relativistic effects diminish.