# Relativistic Spin-0 Feshbach-Villars Equations for Polynomial Potentials

## TL;DR

This paper develops a method to solve relativistic spin-0 particle equations using the Feshbach-Villars formalism, expressing it as an integral equation in the Coulomb-Sturmian basis, and calculating the Green's operator via matrix continued fractions.

## Contribution

It introduces a novel integral equation approach with matrix continued fractions for the Feshbach-Villars formalism applied to polynomial potentials.

## Key findings

- Green's operator computed as matrix continued fraction.
- Method handles Coulomb and linear confinement potentials.
- Analytic continuation ensures convergence at resonant energies.

## Abstract

We propose a solution method for studying relativistic spin-$0$ particles. We adopt the Feshbach-Villars formalism of the Klein-Gordon equation and express the formalism in an integral equation form. The integral equation is represented in the Coulomb-Sturmian basis. The corresponding Green's operator with Coulomb and linear confinement potential can be calculated as a matrix continued fraction. We consider Coulomb plus short range vector potential for bound and resonant states and linear confining scalar potentials for bound states. The continued fraction is naturally divergent at resonant state energies, but we made it convergent by an appropriate analytic continuation.