A note about the factorization of the angular part of the Laplacian and its application to the time-independent Schr\"odinger equation

Abstract

Removing al least one point from the unit sphere in $ R^{3}$ allows to factorize the angular part of the laplacian with a Cauchy-Riemann type operator. Solutions to this operator define a complex algebra of potential functions. A family of these solutions is shown to be normalizable on the sphere so it is possible to construct associate solutions for every radial solution to the time-independant Schr\"odinger equation with a radial potential, such that this family of solutions is square integrable in $R^{3}$. While this family of associated solutions are singular on at least one half-plane, they are square-integrable in almost all of $R^{3}$.