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.

中文翻译：

---

多项式势的相对论 Spin-0 Feshbach-Villars 方程

我们提出了一种研究相对论自旋$0$粒子的求解方法。我们采用 Klein-Gordon 方程的 Feshbach-Villars 形式主义，并以积分方程形式表达形式主义。积分方程以库仑-斯图尔姆基础表示。具有库仑和线性限制势的相应格林算子可以计算为矩阵连分数。我们考虑束缚态和共振态的库仑加短程矢量势和束缚态的线性限制标量势。连分数在共振态能量下自然发散，但我们通过适当的解析延拓使其收敛。