Generators of Gaussian QMS have a GKLS form with noise operators that are linear in creation and annihilation operators, and Hamiltonian which is quadratic. We characterize Gaussian QMS arising in the weak coupling limit (WCL) of a system interacting with some reservoirs. It turns out that their generators, under some natural assumptions, have a GKLS representation with noise operators that are either annihilation or creation operators and Hamiltonian H which is diagonalizable as a quadratic operator.
Bolaños-Servín et al. (Wed,) studied this question.