The Fock representation of the Coulomb interaction. I. Two-body problem

Abstract: The unscreened Coulomb interaction is exactly separable in momentum space at negative energies, by the stereographic projection of Fock, with analytic form factors and strengths. It is derived here as an operator identity for the potential and added, without screening or fitting, as a block of separable terms to the separable nuclear input of a charged pair. The representation is compared with the screening--renormalization method and with Coulomb-distorted form factors on two-body benchmarks: the off-shell $t$~matrix, the Coulomb displacement of the $αp$ Pauli-forbidden state, and the Coulomb-modified phases of the $pp$, $αp$ and $αα$ pairs. The Feshbach--Schur projection of the Pauli-forbidden state is shown to require the Coulomb-dressed eigenstate, which the Fock block provides within the same separable problem. At positive energy a finite truncation is a screened Coulomb potential with an analytic, momentum-dependent radius.