Existence of quantum states for Klein-Gordon particles based on exact and approximate scenarios

In the present study, Kummer's eigenvalue spectra from a charged spinless particle located at spherical pseudo-dot of the form r(2) + 1/r(2) is reported. Here, it is shown how confluent hypergeometric functions have principal quantum numbers for considered spatial confinement. To study systematically both constant rest-mass, m(0)c(2 )and spatial-varying mass of the radial distribution m(0)c(2) + S(r), the Klein-Gordon equation is solved under exact case and approximate scenario for a constant mass and variable usage, respectively. The findings related to the relativistic eigenvalues of the Klein-Gordon particle moving in the spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than m(0)c(2) = 1 fm(-1) in exact scenario. Following analysis also shows eigenvalues satisfy the range of E < m(0)c(2) through approximate scenario.