A generalized discrete singular convolution algorithm improved by regularizing singularities for one electron system

A discrete singular convolution (DSC) algorithm was generalized from uniform discretization to nonuniform discretization by introducing a mapping method to regularize the singularities involved. The approach was demonstrated using a radial Schrِdinger equation of a hydrogen atom with a Coulomb-like potential that involves a singularity. The new approach provides accurate eigenenergies with much reduced computational cost. The mapping method has advantages over the conventional regularization method of the discrete variable representation (DVR) and its descendant, the Lagrange-mesh method in that it can be easily realized in computation and is convenient for selecting suitable grid points.