Solutions of Klein-Gordon equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics

We explore the new analytical solutions for both bound and the new masses of mesons of the Klein–Gordon equation with the modified central complex potential, which describes the heavy light Ǭq, (Q = c, q = u/d, s) mesons and the quarkonium system ǭǭ, (q=c, b, s) via the standard Bopp’s shift method and standard perturbation theory.We have obtained the energy eigenvalues of the ground state E(0)nc(a, b, 0, j, l, m) the first excited state E(1)nc(a, b, 1, j, l, m) and pth  the excited state E(p)nc(a, b, p, j, l, m) in terms of the shift energy (ΔEcc(0, j, l, s, m), ΔEcc(1,j,l,s,m) and ΔEcc(p, j, l, s, m) and (E0l, E1l and Enl)) of ordinary relativistic quantum mechanics. In addition to the parabolic cylinder functions, the Gamma function, the discreet atomic quantum numbers (j, l, s, m) the potential parameters (a and b) and the noncommutativity parameters (θ and σ). In the second part of the research, we will apply the obtained results to calculate the new masses of the mentioned previously mesons in the symmetries of the relativistic three-dimensional noncommutative quantum mechanics. Moreover, some important special cases in the context of the symmetries of the relativistic three-dimensional noncommutative quantum mechanics are treated.