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

Abstract 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.