Minimal Velocity Estimates and Soft Mode Bounds for the Massless Spin-Boson Model

We consider generalised versions of the spin-boson model at small coupling. We assume the spin (or atom) to sit at the origin $${0 \in \mathbb{R}^d}$$0∈Rd and the propagation speed vp of free bosons to be constant, i.e. independent of momentum. In particular, the bosons are massless. We prove detailed bounds on the mean number of bosons contained in the ball $${\{| x | \leq v_p t \}}$${|x|≤vpt}. In particular, we prove that, as $${t \to \infty}$$t→∞ , this number tends to an asymptotic value that can be naturally identified as the mean number of bosons bound to the atom in the ground state. Physically, this means that bosons, that are not bound to the atom, are travelling outwards at a speed that is not lower than vp, hence the term ‘minimal velocity estimate’. Additionally, we prove bounds on the number of emitted bosons with low momentum (soft mode bounds). This paper is an extension of our earlier work in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013). Together with the results in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013), the bounds of the present paper suffice to prove asymptotic completeness, as we describe in De Roeck et al. (Asymptotic completeness in the massless spin-boson model, 2012).