Hulthen plus Hellmann potentials are adopted as the quark–antiquark interaction potential for studying the thermodynamic properties and the mass spectra of heavy mesons. The potential was rendered temperature dependent by replacing the screening parameter with Debye mass. We solved the radial Schrödinger equation analytically using the Nikiforov–Uvarov method. The energy eigenvalues and corresponding wave function in terms of Laguerre polynomials were obtained. The present results are applied for calculating the mass of heavy mesons, such as charmonium [ ] and bottomonium [ ], and thermodynamic properties, such as the mean energy, the specific heat, the free energy, and the entropy. Four special cases were considered when these potential parameters were set to zero, resulting in Hellmann potential, Yukawa potential, Coulomb potential, and Hulthen potential, respectively. The present potential provides satisfying results in comparison with experimental data and the work of other researchers.

The Schrödinger equation was solved for the Hulthén plus screened Kratzer Potential (HSKP) via the Nikiforov–Uvarov – functional analysis (NUFA) method. The bound-state energy and wave function for the HSKP were obtained in a closed form by applying the Greene–Aldrich approximation scheme to the inverse-square term. Using the resulting energy equation, we computed the energy spectra for 12 diatomic molecules (CuLi, TiH, VH, TiC, HCl, LiH, H₂, ScH, CO, I₂, N₂, and NO) for various quantum states. To show the accuracy of our procedure, four special cases of the collective potential were obtained, and the results are in excellent agreement with the existing literature.

We solved the Klein–Gordon equation using the Nikiforov–Uvarov (NU) method with Eckart–Hellmann potential (EHP) model with approximation to the centrifugal term. The analytical expression of the ro-vibrational energy spectra in relativistic, non-relativistic and the corresponding normalised wavefunction were obtained. The numerical bound states energy for various screening parameters at different quantum states and ro-vibrational energies of EHP for VH, TiH, NiC, TiC and CuLi diatomicmolecules were computed. We have also obtained the expectation values of $〈r^{-2}〉$, $〈T〉$ and $〈P^2〉$ using Hellmann–Feynman theorem for the selected diatomic molecules. Graphically, we have illustrated the variation of ro-vibrational energies and expectation values of $〈r^{-2}〉$, $〈T〉$ and $〈P^2〉$. Four special cases of EHP where obtained and the results were in excellent agreements with the existing literature. GRAPHICAL ABSTRACT

Trigonometric Rosen-Morse Potential is employed as a mesonic potential interaction. The extended Nikiforov-Uvarov method is used to solve the N-radial Fractional Schrödinger equation analytically. Using the generalized fractional derivative, the energy eigenvalues are obtained in fractional forms. The current findings are used to calculate the masses of mesons such as charmonium, bottomonium, and heavy-light mesons. The current findings are superior to those of other recent studies and show good agreement with experimental data as a result, the fractional parameter is crucial in optimizing meson masses.

In this study, the Schrödinger equation with the Hulthén plus screened Kratzer potentials (HSKP) are solved via the Nikiforov-Uvarov (NU) and the series expansion methods. We obtained the energy equation and the wave function in closed form with Greene-Aldrich approximation via the NU method. The series expansion method was also used to obtain the energy equation of HSKP. Three distinct cases were obtained from the combined potentials. The energy eigenvalues of HSKP for HCl, LiH, H2, and NO diatomic molecules were computed for various quantum states. To test the accuracy of our results, we computed the bound states energy of HCl and LiH, for a special case of Kratzer and screened Kratzer potentials, which are in excellent agreement with the report of other researchers.