Thursday, February 21, 2013

1302.5012 (Wojciech Dybalski et al.)

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

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.
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