On Rayleigh scattering in the massless Nelson model
Marcel Griesemer, Valentin Kussmaul
TL;DR
The paper targets the long-standing problem of AC for Rayleigh scattering in massless non-relativistic QED models by relaxing the required control on emitted soft photons. It introduces an expanded Fock space (Jakšić–Pillet) to incorporate negative-energy, fake bosons and maps the dynamics to an easier single-Fock-space setting via a unitary $\mathcal W$, enabling a Deift-Simon wave operator $W$ construction and a new propagation estimate. A key contribution is replacing the uniform bound on the photon-number $N$ with a weaker, necessary-and-sufficient condition on the $N$-distribution, together with a minimal escape property, to deduce AC on a fixed energy interval $\Delta=[E,\lambda)$. The results generalize prior AC proofs (e.g., Faupin–Sigal) and apply broadly to spin-boson-like models, offering a robust framework for AC without infrared cutoffs and highlighting the essential role of photon-distribution decay in scattering theory for massless quantum fields.
Abstract
Asymptotic completeness of Rayleigh scattering in models of atoms and molecules of non-relativistic QED is expected, but for a proof we still lack sufficient control on the number of emitted soft photons. So far, this obstacle has only been overcome for the spin-boson model. In a general class of models asymptotic completeness holds provided the expectation value of the photon number $N$ remains bounded uniformly in time. This has been established by Faupin and Sigal. We review and simplify their work, and, more importantly, we replace the bound on $N$ by a weaker assumption on the distribution of $N$ that is both necessary and sufficient for asymptotic completeness.