The effects of noncommutativity on the energy spectra for mass-dependent Klein-Gordon and Shrödinger equations with vector quark-antiquark interaction and harmonic oscillator potential: Application to HLM systems

The analytical solutions of the mass-dependent Klien-Gordon equation (KGE) with the improved harmonic oscillator potential and the improved vector quark-antiquark interaction are derived from the symmetries of three-dimensional relativistic noncommutative quantum mechanics (3D-RNCQM symmetries) using Bopp's shift method and perturbation theory. The energy state equations are sensitive to the global parameters characterizing the noncommutativity space-space (Θ, σ) and the potential parameter (E_nl, a,b, c,m₀, k) in addition to the discrete atomic quantum numbers (j; l; s;m). The energy spectra for the mass-dependent Shrödinger equation with improved vector quark-antiquark interaction and harmonic oscillator potential were found using nonrelativistic limit principles. We also used the current findings to determine the heavy-meson masses of charmonium cc̄ and bottomonium bb̄ in both ordinary nonrelativistic quantum mechanics symmetries and three-dimensional nonrelativistic noncommutative 3D-NRNCQM symmetries. Key Words: Klein-Gordon equation; Shrödinger equation; Harmonic oscillator potential; vector quark-antiquark interaction; Noncommutative quantum mechanics; star products. (author)