In this study, the analytical solutions of the Klein–Gordon equation for any l states of the modified effective mass potential under the modified unequal scalar and vector Coulomb–Hulthén potential (MUSVCH-P) are derived by using an approximation method to the centrifugal potential term in the symmetries of relativistic noncommutative three-dimensional real space (RNC: 3D-RS). The new analytical expressions for eigenvalues of the energy spectrum and the new mass of mesons, such as charmonium and bottomonium that have the quark and antiquark flavor, have been estimated by using Bopp’s shift method, and perturbation theory. The energy state equation depends on the global parameters characterizing the noncommutativity space and the potential parameter (v0,s0,v1,s1,m0,m1,δ) in addition to the Gamma function and the discreet atomic quantum numbers (j,l,s,m). The expression for the new energy spectra is applied to obtain the new mass spectra of heavy quarkonium systems (charmonium and bottomonium) in the symmetries of (RNC: 3D-RS). The comparisons show that our theoretical results are in very good agreement with the reported works.

中文翻译：

---

使用改进的离心项近似和 RNCQM 对称性中的 Bopp 位移方法，用于变形有效质量 Klein-Gordon 方程中变形不等标量和矢量 Coulomb-Hulthén 势的重夸克系统

在本研究中，Klein-Gordon 方程的解析解对于任何l通过使用相对论非对易三维实空间（RNC：3D -RS)。采用Bopp位移法和微扰理论估计了具有夸克和反夸克味道的charmonium和bottomium等能谱特征值和介子新质量的新解析表达式。能量状态方程取决于表征非对易空间的全局参数和势参数(v0,s0,v1,s1,米0,米1,δ)除了 Gamma 函数和谨慎的原子量子数(j,l,s,米). 应用新能谱的表达式来获得（RNC：3D-RS）对称性的重夸克系统（charmonium和bottomium）的新质谱。比较表明，我们的理论结果与报道的工作非常吻合。