Analytical solutions for the Klein–Gordon equation with combined exponential type and ring-shaped potentials

In this study, we have successfully obtained the analytical solutions for the Klein–Gordon equation with new proposed a non-central exponential potential \(V\left( r \right) = D\left[ {1 - \sigma_{0} \coth (\alpha r)} \right]^{2} + (\eta_{1} + \eta_{2} \cos \theta )/r^{2} \sin^{2} \theta\) $V\left(r\right)=D{\left[1-{\sigma }_{0}coth(\alpha r)\right]}^2+({\eta }_1+{\eta }_{2}cos\theta )/r^2{sin}^2\theta$ . Our approach involves a proper approximation of the centrifugal term, with \({l}{\prime}\) $l\prime$ representing the generalized orbital angular momentum quantum number, and the utilization of the Nikiforov–Uvarov method. The resulting radial and angular wave functions are expressed in terms of Jacobi polynomials, and the corresponding energy equation is also derived. Our calculations of the eigenvalues for arbitrary quantum numbers demonstrated significant sensitivity to potential parameters and quantum numbers. Additionally, we evaluate the dependence of energy eigenvalues on screening parameter \(\alpha\) $\alpha$ for arbitrary quantum numbers \(n_{r}\) $n_r$ and \(N\) $N$ to establish the accuracy of our findings. Furthermore, we determine the non-relativistic limits of the radial wave function and energy equation, which align with corresponding previous results in the case where \(\eta_{1} = \eta_{2} = 0\) ${\eta }_1={\eta }_2=0$ .