Two Examples of Quantum Dynamical Semigroups

The Hamiltonians of the considered bi-partite systems are of the form [Formula: see text] Subindex S corresponds to the observed system and R to the reservoir (the enviroment of S). Two classes of systems are distinguished: the discrete-continuous (D-C) and the continuous-continuous (C-C) models. In both cases resevoir operators MR and HR are of continuous spectrum type. In D-C models the operators HS and QS possess discrete spectra. In C-C models, the operators HS and QS are of continuous spectrum type. In each case of our examples the semigroup property for the reduced dynamics of system S is obtained under particular circumstances, depending on the diagonal of the density matrix of the reference state for R. In D-C models, due to discrete spectrum of QS, the semigroup property of the reduced dynamics of the reservoir R is shown to be impossible, unless the coupling to S is trivial.