Solution of the Lippmann–Schwinger Equation for a Partial Wave Transition Matrix with Repulsive Coulomb Interaction

Abstract

A special case where the Lippmann-Schwinger integral equation for the partial wave two-body Coulomb transition matrix for likely charged particles with a negative energy has an analytical solution has been considered. Analytical expressions for the partial- ,-, and-wave Coulomb transition matrices for repulsively interacting particles at the ground-state energy have been derived, by using the Fock method of stereographic projection of the momentum space onto a four-dimensional unit sphere.

Partial wave off-shell Coulomb amplitudes at excited-state energy

“…In the case of the repulsive Coulomb potential analytical expressions for the partial s-, p-and d-wave transition matrices at the energy E = E 1 (γ = 1) have been obtained in Ref. [21].…”

“…This ensures a possibility to obtain the analytical expression for the partial wave Coulomb transition matrices at the energy E 1 for both the attractive [20] and repulsive [21] Coulomb interactions.…”

“…The expressions for three-dimensional Coulomb transition matrix with the explicit removal of the singularities in the transfermomentum variable and the energy have been derived in the papers [15] (for the negative values of the energy E < 0) and [16] (for zeroth and positive energies E ≥ 0). The possibility of obtaining the expressions for partial wave two-body Coulomb transition matrix at the energy of the ground bound state in the analytical form was investigated in previous papers [20] ( in the case of the attractive interaction) and [21] (in the case of the repulsive interaction ).…”

Analytical solutions for three-dimensional Coulomb transition matrix at negative energy and integer values of interaction parameter

“…In specific cases, the analytical solution of the Lippmann-Schwinger equation for the partial two-body Coulomb transition matrices has been realizeed by us previously using the symmetry of the Coulomb systems in the Fock four-dimensional space -in [8,9] (the system with unlike charges) and in [10] (the system with like charges).…”

“…The possibility of the derivation of a simple analytical expression for the partial Coulomb transition matrix, which obey the Lippman-Schwinger equation with the energy of the ground bound state of the two-body complex, has been firstly demonstrated in the papers [8] (in the case of the attractive interaction with the Coulomb parameter γ = −1) and [9] (in the case of the repulsive interaction with the parameter γ = 1). The analytical expressions for the two-particle partial Coulomb transition matrices at the energy of the first-excited state (γ = 2 and γ = −2) have been obtained in [10].…”