Coulomb scattering in the massless Nelson model II. Regularity of ground states

Abstract

For the massless Nelson model we provide detailed information about the dependence of the normalized ground states ${\check{\psi }}_{P,\sigma }$ of the fiber single-electron Hamiltonians $H_{P,\sigma }$ on the total momentum $P$ and the infrared cut-off $\sigma$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \[ \|\partial_{P_i}\checkψ_{P,σ}\|,\ \| \partial_{P_i} \partial_{P_j}\checkψ_{P,σ} \|\leq \frac{c}{σ^{δ_{λ_0}}}, \] for some constant $c$ and a function of the maximal admissible coupling constant ${\lambda }_0\mapsto {\delta }_{{\lambda }_0}$ s.t. ${\lim }_{{\lambda }_0\to 0}{\delta }_{{\lambda }_0}=0$. These results hold both in the infrared-regular and infrared-singular case. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of Haag-Ruelle scattering theory. They should also be relevant for the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.