Effect of System Potential on Quantum Master Equation The usual microscopic derivation of master equation is done with the total Hamiltonian being the sum of the system Hamiltonian, environment and the coupling one. Suppose, the system itself is in a potential, V(x), how does the master equation change? Suppose,as an example, say one wants the master equation of a quantum Brownian particle in a potential which interacts with a suitable bath. 
Does the potential V(x) change the structure of the master equation?
 A: In general, changing the system Hamiltonian by adding a potential will change the structure of the master equation. Of course, the potential $V$ will always enter the coherent Hamiltonian part of the generator $-i[H_0,\rho] \to -i[H_0+V,\rho]$, but it may also change the dissipative part.
However, in the case of quantum Brownian motion, one often considers a heavy Brownian particle whose intrinsic motion is the slowest time scale in the problem (see, e.g., Breuer and Petruccione's textbook). Then the system Hamiltonian is only treated perturbatively. If one also assumes, as usual, that the position of the Brownian particle is coupled to the bath, then the system-bath interaction Hamiltonian commutes with the potential. In that case, the addition of a potential will not change the dissipator to first order in the system Hamiltonian.
In other cases, the intrinsic system dynamics may be quite fast, so that the system Hamiltonian is not a perturbation. Then, the so-called quantum optical master equation may be applicable. This equation describes transitions between the eigenstates of the system Hamiltonian. Therefore, modifying these eigenstates by adding a potential will generally change the structure of the dissipator completely.
