The Derivation of Nonlinear Schrödinger Equations From Quantum Many-Body Evolution: I

The rigorous justication of mean-field type equations (Boltzmann, Vlasov-Poisson, nonlinear Schrödinger...) from the many-body systems they are supposed to describe is a vast and fundamental subject. Since the Nobel Prize winning first observation of Bose-Einstein condensation (BEC), the investigation of this new state of matter and thus the nonlinear Schrödinger equation has become a research hot spot and seen vast developments in recent years. In this talk, we survey the mathematical achievements in the field and discuss the most recent advancement which is the proof of such derivations in the energy-critical setting.