From Quantum Many-Body Dynamics to Energy-Critical Nonlinear Schrodinger Equations and Back

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The project addresses the rigorous justification of mean-field limits of quantum many-body dynamics. Many-body systems arise naturally as fundamental models in physical systems. The simulation of such systems is only possible via some approximations, the so-called mean-field limits, since many-body systems can contain an enormous number of particles. The mathematical justification of mean-field limits, starting from the many-body systems they are supposed to describe, is therefore an issue of fundamental scientific importance. The investigator will address questions arising from the study of Bose-Einstein condensation (BEC), the state of matter of a diluted gas of bosons cooled to temperatures very close to absolute zero. In BEC, a large fraction of the bosons occupies the same quantum state, at which point quantum effects become apparent at the macroscopic scale. Since the first observation of BEC in 1995 by Cornell and Ketterle (who earned the Nobel Prize for their discovery), the investigation of this new state of matter has become one of the most active areas of contemporary research.The focus of this research project is to investigate several problems concerning the fine properties of solutions of the time-dependent many-body Schrödinger equation, in the limit as the particle number tends to infinity at the energy-critical level. This project encompasses three broad directions. The first direction aims to prove that the energy-critical Nonlinear Schrödinger equation is the mean-field limit of quantum many-body dynamics under the Gross-Pitaevskii scaling in the important three-dimensional quintic case. The second direction focuses on space-time regularity of solutions to the many-body Schrödinger equation with focusing interactions. The third direction turns to the study of probability aspects of discrete quantum many-body dynamics. The PI and collaborators will use techniques from harmonic analysis, probability, and spectral theory to analyze these problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.