PDE Seminar

Speaker: 
Jacky Chong (University of Maryland College Park)
Topic: 
"A Dynamical Approximation of Interacting Bosons"

Abstract:
Since the first realization of Bose-Einstein condensate (BEC) in atomic gases over two decades ago, the study of many-body boson systems has gained significant attention in both the physics and math communities. Shortly after the discovery, Lieb and collaborators proved the existence of BEC in a dilute trapped gas at the absolute zero temperature.Later, Erdos, Schlein, and Yau gave a qualitative proof of the persistence of condensate under time evolution and showed that the dynamics of the condensate, in the absence of quantum fluctuations, is well approximated by the Gross-Pitaevskii equation. Despite the success founded in the mean-field theory, recent experiments suggest that mean-field dynamics may not account for the depletion of the condensate, the phenomenon where particles in the condensate escape to higher energy states. Thus, this prompts the question: what lies beyond the mean-field approximation?
     In this talk, we present a rigorous formulation of Bogoliubov theory for interacting bosons and use it to derive the time-dependent Hartree-Fock-Bogoliubov equations (a system of nonlinear Schroedinger-type equations). We will then give a brief sketch of the proof of the well-posedness of the system in 3D and show how the system can be used to get an approximation of the dynamics of the many-body boson system in Fock space. If time permits, we will also discuss some open problems in the field.

Event Date: 
November 14, 2018 - 3:30pm to 4:30pm
Location: 
150 SH
Calendar Category: 
Seminar