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In this work, we obtain the Schr\"odinger equation solutions for the Kratzerpotential plus screened Coulomb potential model using the series expansionmethod. The energy eigenvalues is obtained in non-relativistic regime and thecorresponding unnormalized eigenfunction. Three special cases were obtained. Weapplied the present results to calculate heavy-meson masses of charmonium andbottomonium , and we got the numerical values for 1S, 2S, 1P,2P, 3S, 4S ,1D,2Dand 1F states. The results are in good agreement with experimental data and thework of other researchers.

Analytical solutions of the Schr\"odinger equation with Kratzer-screened Coulomb potential to the Quarkonium systems