Lecturer: Peter Pickl
(LMU Munich) venue and schedule: daily (Monday to Friday), see schedule below start: Monday 29 February 2016 end: Friday 11 March 2016 duration: 20 hours (1 cycle) Synopsis: An extremely topical problem in contemporary mathphys is to obtain rigorous derivations of effective equations for the time evolution of manybody quantum systems constituted by a large number of identical bosons or fermions. There is no hope to solve analytically or numerically the Nbody 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 manybody state that is highly uncorrelated (for example a BoseEinstein 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 manybody dynamics is easily obtained, at least at a formal level, in terms of a onebody orbital that evolves in time according to a nonlinear Schrödinger equation  nonlinearity being the signature of the effective selfinteraction among particles. From a mathphys 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 (multicomponent BoseEinstein condensates, charged particles in interaction with their radiation field, etc.). Prerequisites: A first (undergraduatelike) exposition to the general framework of Quantum Mechanics and some very basic knowledge of linear algebra and functional analysis. Further prerequisites will be given in the 60h course "Mathematical Quantum Mechanics" that precedes this course. Schedule:
Literature: P. Pickl, "Derivation of the time dependent GrossPitaevskii equation with external fields", Rev. Math Phys. 27 (2015), no. 1., 1550003 V. Matulevicius, P. Pickl, "Mean field limits for photons  a way to establish the semiclassical Schrödinger equation", XVIIth ICMP, 567574, World.Sci. Publ. (2014) P. Pickl "A simple derivation of mean field limits for quantum systems", Lett. Math. Phys. 97 (2011), 151164 P. Pickl "Derivation of the time dependent GrossPitaevskii equation without positivity condition on the interaction", J. Stat. Phys. 140 (2010), 7689 A. Knowles, P. Pickl, "Meanfield dynamics: singular potentials and rate of convergence", Comm. Math. Phys. 298 (2010), 101138 N. Benedikter, M. Porta, B. Schlein, "Effective evolution equations from quantum dynamics", Springer Briefs in Mathematical Physics, vol. 7. Springer, Cham (2016) 

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