A note about the factorization of the angular part of the Laplacian and its application to the time-independent Schrödinger equation

2007/05/01 by Daniel Alayon-Solarz, Alayon-Solarz, Daniel
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.CV #math.MP

paper · pdf · doi:10.48550/arxiv.0705.0071

4 pages

arxiv created 2007/05/04 · arxiv updated 2009/12/01

Abstract

Removing al least one point from the unit sphere in R³ 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 Schrödinger equation with a radial potential, such that this family of solutions is square integrable in R³. While this family of associated solutions are singular on at least one half-plane, they are square-integrable in almost all of R³.

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