Effect of the non-commutativity of space on the improved Mobius square plus generalized Yukawa potentials of the Klein–Gordon and Schrödinger equations in 3D-RNCQS and 3D-NRNCQS symmetries

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N 2 , K 2 , NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text], that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.