View on N-dimensional spherical harmonics from the quantum mechanical P\"oschl-Teller potential well

TL;DR

This paper presents a method to derive N-dimensional spherical harmonics using differential equations and special functions, linking them to Schrödinger equations with P"oschl-Teller potentials and Gegenbauer polynomials.

Contribution

It introduces a novel approach connecting spherical harmonics to Schrödinger equations with P"oschl-Teller potentials, providing explicit solutions and eigenvalues.

Findings

Eigenvalues of the Laplace-Beltrami operator are obtained.

Spherical harmonics are expressed via Gegenbauer polynomials.

Method offers a pedagogical perspective on N-dimensional harmonics.

Abstract

In this paper we propose an approach of obtaining of N-dimensional spherical harmonics based exclusively on the methods of solutions of differential equations and the use of the special functions properties. We deduce the Laplace-Beltrami operator on the N-sphere, indicate some instructive relations for the metric, and demonstrate the procedure of separation of the variables. We show that the ordinary differential equations for every variable, except one, can be reduced to the Schr\"odinger equation (SE) with the symmetric P\"oschl-Teller (SPT) potential well by means of certain substitutions. We also exhibit that the solutions of SE with SPT potential are expressed in terms of the Gegenbauer polynomials. The eigenvalues of the Laplace-Beltrami operator and the characteristic numbers of the spherical harmonics are obtained with the use of the properties of the spectrum of SE with SPT potential. The spherical harmonics are constructed as a product of the eigenfunctions of SE with SPT potential multiplied by a easily computable factor function and expressed in terms of the Gegenbauer polynomials. The work has a pedagogical character to some extent.