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We point out that the quantum dynamical map of an open quantum system can begenerated by an effective Liouville operator. The effective Liouville shows thedynamical breaking of time reversibility. This breaking of reversibility isexpressed by the effective Liouville's discrete spectrum having negativeimaginary parts. This generated dynamics of open quantum systems is capable ofmemory effects described by a frequency dependence of the spectrum. When memoryeffects can be neglected or smoothed out, the effective Liouville generates thewell known semi-group dynamics of open systems in the Markov approximation. Thespectral analysis of the effective Liouville -with or without memory- allows torepresent the quantum dynamical map and expectation values of physicalquantities by metric expressions using complex eigenvalues and bi-orthonormaleigenmodes. The long time dynamics is dominated by the zero-mode, which formsthe stationary state of the open system. The relaxation time scale $\tau$ isset by the inverse of the smallest negative imaginary part of the non-vanishingeffective eigenvalues of the effective Liouville. In generic cases ofsystem-environment-coupling this time scale $\tau$ is the relevant scale forboth, the relaxation of diagonal elements (populations) and the decay ofoff-diagonal elements (coherences) of an initial density matrix in thestationary state's diagonal representation. It also sets the time scale afterwhich the entropy tends to its stationary value. The effective Liouville andits spectral content thus give a framework to study the dynamic breaking oftime reversibility, memory effects, decoherence, relaxation to a stationarystate and the role of entropy in open quantum systems.

On Generated Dynamics for Open Quantum Systems: Spectral Analysis of Effective Liouville