Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations

Understand

Let $H_{P,\sigma}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $\sigma>0$.

• We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,\sigma}$ as functions of $P$ and $\sigma$.
• In particular, we show that \[ |\partial_{P^j}f^{n}_{P,\sigma}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,\sigma}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(c\lambda_0)^n}{\sigma^{\delta_{\lambda_0}}} \prod_{i=1}^n\frac{ \chi_{[\sigma,\kappa)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is a numerical constant, $\lambda_0\mapsto \delta_{\lambda_0}$ is a positive function of the maximal admissible coupling constant which satisfies $\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$ and $\chi_{[\sigma,\kappa)}$ is the (approximate) characteristic function of the energy region between the infrared cut-off $\sigma$ and the ultraviolet cut-off $\kappa$.
• While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy.