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Keywords:

  • Coulomb problem;
  • momentum representation;
  • integral equations;
  • Gegenbauer polynomials;
  • Legendre functions.

Abstract

The Schrödinger–Coulomb problem in ℝN, N ≥ 2, is considered in the momentum representation. The adjoint Sturmian eigenvalue problem is discussed first. The resulting radial integral equation is solved with the aid of a symmetric Poisson-type series expansion of the Legendre function of the second kind into products of the Gegenbauer polynomials, established in the 1950's by Ossicini. A relationship between solutions to the Sturmian problem and to an energy eigenvalue problem is then exploited to find the Coulomb bound-state energy levels in ℝN, together with explicit representations of the associated momentum-space wave functions.