Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering

We construct two-electron scattering states and verify their tensor product struc- ture in the infrared-regular massless Nelson model. The proof follows the lines of Haag- Ruelle scattering theory: Scattering state approximants are defined with the help of two time-dependent renormalized creation operators of the electrons acting on the vacuum. They depend on the ground state wave functions of the (single-electron) fiber Hamiltonians with infrared cut-off. The convergence of these approximants as t → ∞ is shown with the help of Cook’s method combined with a non-stationary phase argument. The removal of the infrared cut-off in the limit t → ∞ requires sharp estimates on the derivatives of these ground state wave functions w.r.t. electron and photon momenta, with mild dependence on the infrared cut-off. These key estimates, which carry information about the localization of the electrons in space, are obtained in a companion paper with the help of iterative analytic perturbation theory. Our results hold in the weak coupling regime.