Generalized Functional Analysis Approach for Bound State Solutions with Solvable and Quasi-Solvable Potentials

In this paper, a generalized functional analysis approach has been used to obtain the bound state solutions of the Schrödinger equation with solvable and quasi-solvable potentials. We extended the Nikiforov-Uvarov functional analysis method which has been limited to obtaining bound states solutions of exponential-type potentials. Here, we have generalized the approach to obtain the approximate and exact energy spectra and the corresponding wave functions of the Schrödinger equation under the Coulomb, Pseudo harmonic, hyperbolic, q-deformed hyperbolic and perturbed Coulomb potentials in closed forms. These potential functions with the Schrödinger equation were transformed into standard second-order linear differential equations such as the Whittaker, associated Laguerre, Gauss hypergeometric and the confluent Heun differential equations. The parameters of the wave functions and energy quantization conditions were derived from a direct comparison with the quantized systems of differential equations. Our results are in excellent agreement with the earlier works in which different approaches were utilized. Also, the quasi-exact energies for fixed potential parameters were found to be in good agreement with energy eigenvalues obtained via the matrix Numerov method.