  For the model of a multi-level system in the rotating wave approximation we
obtain the corrections for a usual weak coupling limit dynamics by means of
perturbation theory with Bogolubov-van Hove scaling. It generalizes our
previous results on a spin-boson model in the rotating wave approximation.
Additionally, in this work we take into account some dependence of the system
Hamiltonian on the small parameter. We show that the dynamics is long-time
Markovian, i.e. after the bath correlation time all the non-Markovianity could
be captured by the renormalization of initial condition and correlation
functions.
