Speaker: 李用声 教授 (华南理工大学)
Title: Derivation of the Nonlinear Schrodinger Equation From Quantum Many-Body Dynamics
Time: 2019年9月25日，下午3:30-5:30
Location: 数学中心813
Abstract:
In this talk, the quantum many-body dynamics with m-body interactions in Λ = Rd or Λ = Td is considered.
This problem arises from the study of Bose-Einstein condensate (BEC). A Bose- Einstein condensate (BEC) is a peculiar gaseous state, a state of matter of a dilute gas of low densities (called bosons) cooled to temperatures very close to absolute zero (-273.15 ◦C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. This state of matter can be used to explore fundamen- tal questions in quantum mechanics, such as the emergence of interference, decoherence, superfluidity and quantized vortices. Investigating various condensates has become one of the most active areas of contemporary research. A single particle in quantum mechanics is governed by a linear one-body Schro ̈dinger equation. If N bodies interact quantum mechanically, they are governed by the N-body linear Schro ̈dinger equation due to the superposition principle. Thus one arrive at the N-particle dynamic system for the anal- ysis of BEC, which is impossible to solve or simulate when N is large. Therefore, it is necessary to find reductions or approximations. It is long believed that NLS is the mean- field limit for these N-body systems. This has to be studied via rigorous mathematical proofs.
In this talk the main idea of this theory is introduced. The derivation of a certain type of nonlinear Schr ̈odinger equation from this model has been established under certain assumptions. The main methods are the convergence of the Bogoliubov-Born-Green- Kirkwood-Yvon (BBGKY) hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and the uniqueness of the solutions to the GP hierarchy. In the proof of convergence the sharp trace estimate plays a fundamental role. For the large solution uniqueness, some hierarchical uniform frequency localization (HUFL) property for the GP hierarchy is used.