The Klein-Gordon (KG) equation is one of the fundamental relativistic wave equation that describes the motion of spin zero particles.
[14] Remarkable efforts have been executed to examine the solutions of the KG equation with a various number of potential energies. Yi
et al. employed RM type vector and scalar potential energies to obtain the s-wave bound state energy spectra.
[15] Villalba
et al. examined the bound state solution of a spatially one-dimensional cusp potential energy in the KG equation.
[16] Olgar
et al. employed a supersymmetric technique to obtain a bound state solution of the s-wave KG equation with equal scalar and vector Eckart type potential energy.
[17] Only two years later, they applied the asymptotic interaction method (AIM), which is originally introduced by Ciftci
et al.,
[18] to calculate an energy spectrum of the s-wave KG equation with the mixed scalar and vector generalized Hulth$\acute{\rm e}$n potential in one dimension.
[19] Then, he used AIM to investigate bound state solution of three different potential energies, namely linear, Morse and Kratzer, in the KG equation.
[20] In 2010, Xu
et al. studied the bound state solution of the KG equation with mixed vector and scalar PT potential energy with a non zero angular momentum parameter.
[21] Ikot
et al. obtained an exact solution of the Hylleraas potential energy in the KG equation.
[22] Jia
et al. examined the bound state solution of the KG equation with an improved version of the MR potential energy.
[23] Hou
et al. studied the bound state solution of the s-wave KG equation with vector and scalar WS potential energy.
[24] Rojas
et al. used the vector WS barrier in the KG equation and presented the continuum state solution.
[25] Later, Hassanabadi extended that study with an addition of scalar WS potential energy term.
[26] Arda
et al. employed Nikiforov-Uvarov (NU) and studied the modified WS potential energy with position dependent mass in the KG equation in three dimensions.
[27] Badalov
et al. used NU and Pekeris approximation to study any $l$ state of the KG equation.
[28] Bayrak
et al. investigated the generalized WS potential energy in the KG equation for zero
[29] and non-zero
[30] values of the angular momentum parameter. One of the authors of this manuscript, Lütfüoğlu, with his collaborators examined the mixed vector and scalar generalized symmetric WS potential energies for the scattering case in the KG equation first under the equal magnitudes and signs (EMES), and then, in the equal magnitudes and opposite signs (EMOS).
[31] Later, he investigated the same problem in the bound state case.
[32] Beside these studies, multi-parameter exponential type potential energies
[33]-[35] and non central potentials
[36]-[37] are examined in the KG equation.