**Webpage:**

http://people.sissa.it/~alemiche/teaching2016_Manybody_Quantum_Dynamics....

**Synopsis:**

An extremely topical problem in contemporary math-phys is to obtain rigorous derivations of effective equations for the time evolution of many-body quantum systems constituted by a large number of identical bosons or fermions. There is no hope to solve analytically or numerically the N-body Schrödinger equation when N is large (in the applications N ranges from a few thousands to billions or more). If at time t=0 the system is prepared in a many-body state that is highly uncorrelated (for example a Bose-Einstein condensate in the case of bosons), one expects that at later times the system still displays very few correlations among particles (this is the so called "propagation of chaos"). With this Ansatz a much simpler description of the many-body dynamics is easily obtained, at least at a formal level, in terms of a one-body orbital that evolves in time according to a non-linear Schrödinger equation -- non-linearity being the signature of the effective self-interaction among particles. From a math-phys viewpoint the interest is to make such a formal derivation mathematically rigorous in the limit of infinite N, supplementing it with additional information such as the approximation error for large but finite N or the short/long time behaviour. As the course will show, this has stimulated the development of several new and alternative techniques of algebraic and analytical nature, as well as the quest for effective evolution equations for more complex systems (multi-component Bose-Einstein condensates, charged particles in interaction with their radiation field, etc.).

The course will end with a special AMPQ seminar by Nikolai Leopold (LMU Munich), who will be lecturing on a very recent application of Pickl's methods on Mean-field limits of charged particles in interaction with their radiation field