Lecturer: Peter Pickl
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 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.).
Pre-requisites: A first (undergraduate-like) exposition to the general framework of Quantum Mechanics and some very basic knowledge of linear algebra and functional analysis. Further pre-requisites will be given in the 60h course "Mathematical Quantum Mechanics" that precedes this course.
P. Pickl, "Derivation of the time dependent Gross-Pitaevskii 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", XVII-th ICMP, 567-574, World.Sci. Publ. (2014)
P. Pickl "A simple derivation of mean field limits for quantum systems", Lett. Math. Phys. 97 (2011), 151-164
P. Pickl "Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction", J. Stat. Phys. 140 (2010), 76-89
A. Knowles, P. Pickl, "Mean-field dynamics: singular potentials and rate of convergence", Comm. Math. Phys. 298 (2010), 101-138
N. Benedikter, M. Porta, B. Schlein, "Effective evolution equations from quantum dynamics", Springer Briefs in Mathematical Physics, vol. 7. Springer, Cham (2016)