Sums of Products of Coulomb Wave Function
over Degenerate States

M. I. Chibisov^1, *, A. M. Ermolaev², M. Sherkani³, and F. Bruiar⁴

¹Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia

²Physique Théorique, Faculté des Sciences CP 227, Université Libre de Bruxelles, B-1050 Bruxelles, Belgium

³UFR Physique Corpusculaire, FST Fes-Saiss, Morocco

⁴Départment de Physique, Unité Fyam, Bâtiment Marc de Hemptinne, B-1348, Louvain-la-Neuve, Belgium

*e-mail: chib@qq.nfi.kiae.su

Received August 19, 1999

Abstract—The sums of products of Coulomb wave function over degenerate states are expressed in terms of qua-
dratic forms that depend on the wave function of only one state with zero orbital angular momentum l = m = 0.
These sums are encountered in many fields in the physics of atoms and molecules, for example, in investigations
of the perturbation of degenerate atomic energy levels of a small potential well, a delta-function potential. The
sums were found in an investigation of the limit of the Coulomb Green’s function G(r, r', E), where the energy
parameter E approaches an atomic energy level: E E_n, E_n = –Z ²/2n². The Green’s function found by
L.Hostler and R. Pratt in 1963 was used. The result obtained is a consequence of the degeneracy of the
Coulomb energy levels, which in turn is due to the four-dimensional symmetry of the Coulomb problem.
© 2000 MAIK “Nauka/Interperiodica”.

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