Approximate Solutions to the Klein-Fock-Gordon Equation for the Sum of Coulomb and Ring-Shaped-Like Potentials

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass M, described by
the Klein-Fock-Gordon equation with equal scalar Sðr
!Þ and vector Vðr
!Þ Coulomb plus ring-shaped potentials. It is shown that the
system under consideration has both a discrete at jEj < Mc2 and a continuous at jEj > Mc2 energy spectra. We find the analytical
expressions for the corresponding complete wave functions. A dynamical symmetry group SUð1, 1Þ for the radial wave equation
of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic
way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞
go over into the corresponding expressions for the nonrelativistic problem.