Connection formulas between Coulomb wave functions

Abstract

The mathematical relations between the regular Coulomb function F η (ρ) and the irregular Coulomb functions H ± η (ρ) and G η (ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of . These relations, referred to as "connection formulas", form the basis of the theory of Coulomb wave functions, and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function F η (ρ) are studied, in particular under the transformation → − − 1, by means of the modified Coulomb function Φ η (ρ), which is entire in the dimensionless energy η −2 and the angular momentum . Then, it is shown that, for integer or half-integer , the irregular functions H ± η (ρ) and G η (ρ) can be expressed in terms of the derivatives of Φ η, (ρ) and Φ η,− −1 (ρ) with respect to . As a consequence, the connection formulas directly lead to the description of the singular structures of H ± η (ρ) and G η (ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η −1 .

Complex-energy analysis of proton-proton fusion

“…In particular, they tend toward nonzero functions of r/a B at zero energy. The function Φ η, is related to F η, by [32]…”

“…The actual origin of this maximum is revealed by the continuation of Λ to complex energies, as provided by Eq. Furthermore, in contrast to Λ, the function L is holomorphic in the neighborhood of E = 0, because of the properties of the modified Coulomb functions Φ η,0 and Ψ η,0 in the limit E → 0 [18,32]. Therefore, it follows from Eq.…”

“…However, such a method is not accurate due to the non-negligible influence of the interval itself [12,15]. [18,32,38] of the analytic continuation of Λ 2 . The complex phase is highlighted by colors.…”

“…This useful property is preserved by the Laplace transforms (B1) and (B2). Therefore, the function ψ β, ( ) can be calculated from derivatives of φ β, ( ) as follows [32] ψ β, w η φ β,…”

On radial Schrödinger operators with a Coulomb potential: general boundary conditions

“…1. See also [4,9,[12][13][14][15][16][17][18][24][25][26][27][28] and references therein. However, essentially all these references are devoted to real parameters b; m and self-adjoint realizations of Whittaker operators.…”