Solution of the momentum-space Schrödinger equation for bound states of the N-dimensional Coulomb problem (revisited)

Abstract

The Schrödinger-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 Schrödinger-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 Schrödinger-Coulomb bound-state energy levels in ${\mathbb{R}}^N$, together with explicit representations of the associated normalized momentum-space Schrödinger-Coulomb Hamiltonian eigenfunctions.

LaTeX2e, 13 pages; some improvements made; references added

References in corpus (2)

• The Coulomb Green's Function in Two Dimensions
• Multipole expansions and Fock symmetry of the Hydrogen atom

Cited by in corpus (6)

• Deviations of the exciton level spectrum in cuprous oxide from the hydrogen series
• Interaction of charged impurities and Rydberg excitons in cuprous oxide
• The phonon assisted absorption of excitons in Cu${}_2$O
• Waveguides for Rydberg excitons in Cu${}_{\text{2}}$O from strain traps
• Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D
• Wave functions of the Hydrogen atom in the momentum representation