Solution of the momentum-space Schr\"odinger equation for bound states of the N-dimensional Coulomb problem (revisited)

The Schr"odinger-Coulomb Sturmian problem in ${\mathbb{R}}^N$, $N⩾2$, is considered in the momentum representation. An integral formula for the Gegenbauer polynomials, found recently by Cohl [arXiv:1105.2735], is used to separate out angular variables and reduce an integral Sturmian eigenvalue equation in ${\mathbb{R}}^N$ to a Fredholm one on ${\mathbb{R}}_{+}$. A kernel of the latter equation contains the Legendre function of the second kind. A symmetric Poisson-type series expansion of that function into products of the Gegenbauer polynomials, established by Ossicini [Boll. Un. Mat. Ital. 7 (1952) 315], is then used to determine the Schr"odinger-Coulomb Sturmian eigenvalues and associated momentum-space eigenfunctions. Finally, a relationship existing between solutions to the Sturmian problem and solutions to a (physically more interesting) energy eigenvalue problem is exploited to find the Schr"odinger-Coulomb bound-state energy levels in ${\mathbb{R}}^N$, together with explicit representations of the associated normalized momentum-space Schr"odinger-Coulomb Hamiltonian eigenfunctions.