%0 Journal Article
%T The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II
%A Jos¨¦ F. CariŁżena
%A Manuel F. RaŁżada
%A Mariano Santander
%J Mathematics
%D 2012
%I arXiv
%R 10.1063/1.4757604
%X This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($\kappa>0$) and $H_\k^3$ ($\kappa<0$), to the standard {\itshape spherical waves} in $E^3$. The curvature $\k$ is considered as a parameter and for any $\k$ we show how the radial Schr\"odinger equation can be transformed into a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both $\k$-dependent and $\k$-independent), and are explicitly obtained.
%U http://arxiv.org/abs/1211.2076v1