Summary: Using the concept of conformable fractional derivative, we study the properties of fractional \(N\)-dimensional Schrödinger equation for the potential \(V(r) = - \frac{k}{r} + \frac{g}{r^2} + a r + b r^2\). The extended Nikiforov-Uvarov method is generalized to the fractional domain and then employed to obtain the analytic exact energy eigenvalues and eigenfunctions and their dependence on the fractional order \(\alpha\) and the dimension \(N\). To test its applicability, we apply the method on heavy quarkonia systems, and reproduce their mass spectra and fractional radial probabilities at different values of \(\alpha\) and \(N\). Comparing the mass spectra with the experimental data, we discuss to what extent fractional models can account for some features in the description of heavy quarkonia at certain dimensional space.

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