Perturbed Coulomb Potentials in the Klein–Gordon Equation: Quasi-Exact Solution

Using the Lie algebraic approach, we present the quasi-exact solutions of the relativistic Klein–Gordon equation for perturbed Coulomb potentials namely the Cornell potential, the Kratzer potential and the Killingbeck potential. We calculate the general exact expressions for the energies, corresponding wave functions and the allowed values of the parameters of the potential within the representation space of sl(2) Lie algebra. In addition, we show that the considered equations can be transformed into the Heun’s differential equations and then we reproduce the results using the associated special functions. Also, we study the special case of the Coulomb potential and show that in the non-relativistic limit, the solution of the Klein–Gordon equation converges to that of Schrödinger equation.