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

TL;DR

This paper develops a method to solve the N-dimensional momentum-space Schrödinger-Coulomb bound-state problem using integral formulas for Gegenbauer polynomials and Legendre functions, providing explicit eigenvalues and eigenfunctions.

Contribution

It introduces a novel approach to solving the N-dimensional Coulomb problem in momentum space by reducing the integral eigenvalue equation to a solvable form using special functions.

Findings

Derived explicit eigenvalues for the N-dimensional Coulomb problem.

Obtained normalized momentum-space eigenfunctions explicitly.

Connected Sturmian solutions to energy eigenstates for bound states.

Abstract

The Schr\"odinger-Coulomb Sturmian problem in $\mathbb{R}^{N}$, $N\geqslant2$, 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.