In this paper, we investigate the exact bound state solution of the Klein–Gordon equation for an energy-dependent Coulomb-like vector plus scalar potential energies. To the best of our knowledge, this problem is examined in literature with a constant and position dependent mass functions. As a novelty, we assume a mass-function that depends on energy and position and revisit the problem with the following cases: First, we examine the case where the mixed vector and scalar potential energy possess equal magnitude and equal sign as well as an opposite sign. Then, we study pure scalar and pure vector cases. In each case, we derive an analytic expression of the energy spectrum by employing the asymptotic iteration method. We obtain a nontrivial relation among the tuning parameters which lead the examined problem to a constant mass one. Finally, we calculate the energy spectrum by the Secant method and show that the corresponding unnormalized wave functions satisfy the boundary conditions. We conclude the paper with a comparison of the calculated energy spectra versus tuning parameters.

中文翻译：

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具有能量依赖势的 Klein-Gordon 方程的束缚态解

在本文中，我们研究了 Klein-Gordon 方程的精确束缚态解，用于与能量相关的类库仑向量加标量势能。据我们所知，这个问题在文献中用常数和位置相关的质量函数进行了检查。作为一种新颖性，我们假设一个取决于能量和位置的质量函数，并用以下情况重新审视这个问题：首先，我们检查混合矢量和标量势能具有相等大小和等号以及相反的情况标志。然后，我们研究纯标量和纯向量情况。在每种情况下，我们通过采用渐近迭代法推导出能谱的解析表达式。我们在调整参数之间获得了一个非平凡的关系，这导致所检查的问题成为一个恒定质量的问题。最后，我们通过正割法计算能谱，并证明相应的非归一化波函数满足边界条件。我们通过比较计算的能谱与调谐参数来结束本文。