Two-particle Coulomb Green's function in the momentum space with explicity separated singularities

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

“…With the help of the stereographic projection of the momentum space onto the four-dimensional sphere of the unit radius, a singleparameter integral representation of the three-dimensional Coulomb Green's function has been derived in the papers by Bratsev and Trifonov [4] and Schwinger [5]. With the use of the Fock symmetry, the general expressions for the three-dimensional Coulomb transition matrix with explicitly separated singularities in the transfer momentum and the energy have been derived in the papers [6] (in the case of the negative energy) and [7] (in the case of zero and positive energies).…”

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

“…There are a number of representations for the twobody Coulomb transition matrix [8][9][10][11][12][13][14][15][16]. Of special interest is the study of the Coulomb transition matrix, by taking advantage of the Coulomb system symmetry in the Fock four-dimensional Euclidean space [17].…”

“…Earlier, the Fock method was applied in Bratsev-Trifonov's [10] and Schwinger's [12] works in order to derive the Coulomb Green's function in the one-parameter integral form. Expressions for the three-dimensional Coulomb transition matrix with explicitly singled out transferred momentum and energy singularities were obtained in works [14] (for negative energies, < 0) and [15] (for zero and positive energies, ≥ 0).…”

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

“…Presently, there are known several representations of the two-body Coulomb transition matrix [8][9][10][11][12][13][14][15][16][17][18]. The one-parameter integral representations for the Coulomb Green's function which take imediately into account the existing symmetry of the Coulomb system in Fock's four-dimensional Euclidean space [19] have been firstly obtained in the papers by Bratsev and Trifonov [10] and Schwinger [13].…”

“…The one-parameter integral representations for the Coulomb Green's function which take imediately into account the existing symmetry of the Coulomb system in Fock's four-dimensional Euclidean space [19] have been firstly obtained in the papers by Bratsev and Trifonov [10] and Schwinger [13]. 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 ).…”