Relativistic Spin-0 Feshbach-Villars Equations for Polynomial Potentials

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.