Quantum Markovian master equations and non-Lindbladian generators

We describe an approach to strongly continuous quantum dynamical semigroups via completely positive perturbations of their (in general, unbounded) generators. The semigroup is standard if its generator is "of Lindblad's type", i.e. it is obtained by a completely positive perturbation of a "no-event" generator. Then we consider two cases of dynamical semigroups obtained by singular perturbations of a standard generator. First, we describe an example which gives a positive answer to a conjecture of Arveson concerning possible triviality of the domain algebra. Second, we consider an improved and simplified construction of a nonstandard dynamical semigroup which gives answer to the question on existence of dynamical semigroups with non-Linbladian generators.