An Infinite Level Atom coupled to a Heat Bath

We consider a $W^*$-dynamical system $(\Mg,\taug)$, which models finitely many particles coupled to an infinitely extended heat bath. The energy of the particles can be described by an unbounded operator, which has infinitely many energy levels. We show existence of the dynamics $\taug$ and existence of a $(\beta,\taug)$ -KMS state under very explicit conditions on the strength of the interaction and on the inverse temperature $\beta$.