Lie-admissible perturbation methods for open quantum systems

We consider open quantum systems described by a Hamiltonian of the type H0+λV, where λ is a small parameter. For such systems, we develop perturbative methods of solution of the corresponding Liouville-von Neumann and Schrödinger equations, by introducing “dissipation” operators which connect conservative to dissipative systems. In the case of the density matrix, the corresponding operator Λ is nothing but the non-unitary Λ-transformation of Misra, Prigogine and Courbage. Our perturbative approach possesses a Lie-admissible structure, since the “dissipation” operators obey time-evolution equations whose brackets are the product of a Lie-admissible algebra. Explicit solutions for such operators are found in the form of series expansions in λ. The matrix formulation of the above results is also given.