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

Wojciech Dybalski, Alessandro Pizzo
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 $\|\partial_{P}^{\beta}\check{\psi}_{P,\sigma}\|\leq \frac{c}{\sigma^{\delta_{\lambda_0}}},$ for any multiindex $\beta$, s.t. $0\leq |\beta|\leq 2$ and some function of the maximal admissible coupling constant $\lambda_0\mapsto \delta_{\lambda_0}$ s.t. $\lim_{\lambda_0\to 0}\de_{\lambda_0}=0$. Analogous bounds are obtained for the derivatives w.r.t. $P$ of the $q$-particle momentum wavefunctions $f^q_{P,\sigma}$ of $\check{\psi}_{P,\sigma}$. These results are exploited in a companion paper 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.
View original: http://arxiv.org/abs/1302.5012