Existence of quantum states for Klein-Gordon particles based on exact and approximate scenarios with pseudo-dot spherical confinement

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 spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than $m_{0}=1$ fm$^{-1}$ in exact scenario. Following analysis shows eigenvalues satisfy the range of $E<m_{0}$ through approximate scenario.