Ultraviolet Renormalization of Spin Boson Models with Normal and 2-Nilpotent Interactions

Abstract We study the ultraviolet problem for models of a small quantum mechanical system with bounded Hamiltonian linearly coupled to a bosonic quantum field. In the main examples the small system is finite dimensional, such as in the (many-)spin boson model or its rotating-wave approximation. If the state change of the system upon emission or absorption of a boson is given by a bounded and normal or 2-nilpotent operator, which is the case for the previously named examples, we prove an optimal renormalization result. In particular, we prove the norm resolvent convergence of appropriately regularized models to the renormalized one. Our method consists of a dressing transformation argument in the normal case and an appropriate interior boundary condition for the 2-nilpotent case.