?The Schr?dinger equation describes the evolution of isolated quantum systems. But when subjected to external noises, the dynamical equation of a quantum system becomes entirely different, both from a physical and a mathematical point of view. Such open systems exhibit typically irreversible behaviour in the dynamics. They are found everywhere in nature and are the subject of theoretical and experimental study in Physics, Chemistry, Biology and Mathematics. With rare exceptions, the complexity of the equations of motion prohibits us from finding the exact solution for the dynamics of open systems. One then resorts to plausible simplifying approximations, such as the widely used Born and Markov approximations, which result in a more tractable situation, analytically and numerically. To ensure the accuracy of the approximations with precision is very difficult, however! I will outline some results on the validity of the Born and Markov approximation, based on a mathematically rigorous method, and present applications to concrete models from quantum optics and quantum chemistry.

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