Exactly solvable (ES) systems are those for which the full, discrete spectrum can be solved in closed form. In this work, we argue that a moment’s representation analysis can generate these closed-form expressions for the energy in a more direct and transparent manner than the popular Nikiforov–Uvarov (NU) procedure. NU analysis strips the asymptotic form of the physical states. We retain these to generate appropriate moment equations. We show how the form of these moment equations leads to closed-form energy expressions. The wave functions can then be generated as well. Our analysis is extendable to quasi-exactly solvable systems (QES; those for which a subset of the discrete spectrum can be generated in closed form). Two formulations are presented. One of these affirms that a previously developed, general, moment quantization procedure is exact for ES and QES states. This method is referred to as the orthogonal polynomial projection quantization method. It combines moment equation representations for physical states with weighted polynomial expansions (Handy and Vrinceanu. J. Phys. A: Math. Theor. 46, 135202 (2013). doi:10.1088/1751-8113/46/13/135202 ). We also show that in implementing any numerical search procedure to determine the quantum parameter regimes corresponding to ES or QES states, our procedure is more reliable (i.e., numerically stable) than using a Hill determinant formulation. We develop our formalism, demonstrate its effectiveness, and prove its equivalence to the NU approach for ES systems.
Quantum square-well with logarithmic central spike
