  We focus on a recently developed generalized pseudospectral method for
accurate, efficient treatment of certain central potentials of interest in
various branches in quantum mechanics, usually having singularity. Essentially
this allows optimal, nonuniform spatial discretization of the pertinent
single-particle Schrodinger equation satisfying Dirichlet boundary condition
leading to standard diagonalization of symmetric matrices. Its validity and
feasibility have been demonstrated for a wide range of important potentials
such as Hulth\'en, Yukawa, generalized spiked harmonic oscillators, Hellmann,
Coulomb potentials without/with various perturbations (for instance, linear and
quadratic) etc. Although initially designed for singular potentials, this has
also been remarkably successful for various other cases such as power-law,
logarithmic, harmonic potentials containing higher order perturbations, 3D
rational potentials as well as confinement studies. Furthermore, a large number
of low-, moderately high-, high-lying multiply excited Rydberg states such as
singly, doubly excited He as well as triply excited hollow $2l2l'2l'' (n \ge
2)$ and $3l3l'3l''$ doubly-hollow resonances in many-electron atoms have been
treated by this approach within a KS DFT with great success. This offers very
high-quality results for both ground and higher lying states for arbitrary
values of potential parameters (covering both weak and strong coupling) with
equal ease and efficacy. In all cases, excellent agreement with literature
results are observed; in many cases this surpasses the accuracy of all other
existing results while in other occasions our results are comparable to the
best ones available in literature.
