Bound State Solution of the Klein-Fock-Gordon equation with the Hulthén plus a Ring-Shaped like potential within SUSY quantum mechanics

Abstract: In this paper, the bound state solution of the modified Klein-Fock-Gordon equation is obtained for the Hulth\'en plus ring-shaped lake potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any $l\neq0$ angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein-Fock-Gordon equation of the Hulth\'en plus ring-shaped like potential are obtained by Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary $l$ states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to $n_r$ radial and $l$ orbital quantum numbers.