Transience and recurrence of quantum Markov semigroups
The authors introduce the notion of a potential for a quantum Markov semigroup, and classes of transient and recurrent quantum Markov semigroups. It is shown that an irreducible semigroup is either transient or recurrent. Transient semigroups are characterized in terms of the existence of non-trivial superharmonic operators. As examples, the quantum Brownian motion and the Jaynes-Cummings micromaster model of quantum optics are considered. Applications to scattering theory are given. In particular, it is shown that for a transient quantum semigroup \(T_t\) any state \(\varphi\) is scattering, that is for any finite-dimensional projection \(p\), \[ \lim_{t\to \infty}\varphi T_t(p)=\lim_{t\to \infty}T_{*t}(\varphi)(p)=0 . \]
scientific article; zbMATH DE number 1968523