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.Keywords Klein-Gordon equation • Kummer's differential equation • eigenvalues • relativistic particles 1 IntroductionQuantum mechanical wave functions are represented by probability distributions near a certain spatial point which are localized in the interaction field. Based on the spatial motions for quantum mechanical particles-represented by relativistic and nonrelativistic eigenstates-it is important to analyze the discrete energies for quantum systems in the electronic, nuclear and particle physics. In addition to numerous studies, the quantum physical process has been applied to the external field on the electronic-interactions through plasma [1] and condensed matter [2; 3; 4; 5]. Concerning the Klein-Gordon equation, which describes relativistic spin-zero energy levels, it has been shown that the eigenvalue equation leads to spatial confluent hypergeometric functions, not only in the harmonic oscillator [6; 7; 8] , but also in the fractional regime [9]. Furthermore, Mie-formation Shipito Address

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Approximate analytical solutions of the Dirac equation in the case of pseudospin and spin symmetry limits are investigated under the Deng-Fan potential by applying the asymptotic iteration method for the arbitrary quantum numbers n and κ. Some of the numerical results are also represented in both pseudospin symmetry and spin symmetry limits.

The exact solutions of the N-dimensional Klein-Gordon equation in the presence of an exactly solvable potential of V (r) = D e (r/r e − r e /r) 2 type have been obtained. The N dimensional Klein-Gordon equation has been reduced to a first-order differential equation via Laplace transformation. The exact bound state energy eigenvalues and corresponding wave functions for CH, H 2 , and HCl molecules interacting with pseudoharmonic oscillator potential in the arbitrary N dimensions have been determined. Bound state eigenfunctions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric functions.