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

TL;DR

This paper develops a rigorous framework for two-electron scattering in the massless Nelson model, establishing the structure of scattering states and their convergence using advanced mathematical techniques.

Contribution

It constructs and verifies the tensor product structure of two-electron scattering states in the infrared-regular massless Nelson model, extending Haag-Ruelle theory.

Findings

Successful construction of two-electron scattering states

Proof of tensor product structure of scattering states

Convergence established using Cook's method and phase analysis

Abstract

We construct two-electron scattering states and verify their tensor product structure 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 ground state wave functions of the (single-electron) fiber Hamiltonians with infrared cut-off. Convergence of these approximants as $t\to \infty$ is shown with the help of Cook's method combined with a non-stationary phase argument. Removal of the infrared cut-off in the limit $t\to \infty$ 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 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.