Mathematical Analysis of Dispersion and Transport in Quantum Dynamics

本研究解决了由相互作用的量子力学粒子（玻色子或费米子类型）组成的大型物理系统中出现的问题，并用严格的数学方法进行了分析。PI将研究推动重量子示踪粒子进入玻色子气体的效果，并研究玻色-爱因斯坦凝聚现象发生时量子涨落的动力学行为。此外，PI将研究电子（费米子）在无序介质中的运动，描述半导体等系统。在另一项研究中，PI和他的合作者最近引入了一种方法，用量子力学的方法来研究描述经典气体和流体的方程，使用一种被称为维格纳变换的数学装置；该方法将在本项目中得到进一步发展。这些项目包括研究生和博士后研究人员的培训机会。PI将在Hartree-Fock-Bogoliubov近似及更高水平上研究玻色-爱因斯坦凝聚体周围的涨落动力学，使用非线性偏微分方程和量子场论中的重整化方法的组合。此外，PI将在具有平均场相互作用的弱无序多体安德森模型的背景下，研究费米子系统在随机介质中传播的有效方程的推导。经典玻尔兹曼方程的维格纳变换分析和色散非线性偏微分方程的方法将得到进一步发展。分析这些问题的一个重要成果是发展了新的数学技术，将非线性偏微分方程、谐波分析和量子场论的方法结合起来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research addresses problems emerging from the physics of large systems of interacting quantum mechanical particles (of boson or fermion type), analyzed with rigorous mathematical methods. The PI will study the effects of impelling a heavy quantum tracer particle into a gas of bosons, and investigate the dynamical behavior of quantum fluctuations when the phenomenon of Bose-Einstein condensation occurs. In addition, the PI will study the motion of electrons (which are fermions) in disordered media, describing systems such as semiconductors. In another line of research, the PI and his collaborators have recently introduced an approach to study equations describing classical gases and fluids with methods in quantum mechanics, using a mathematical device known as the Wigner transform; this method will be further developed in this project. The projects includes training opportunities for graduate students and postdoctoral researchers. The PI will study the fluctuation dynamics around a Bose-Einstein condensate on the level of the Hartree-Fock-Bogoliubov approximation and beyond, using a combination of methods from nonlinear partial differential equations and renormalization in quantum field theory. Moreover, the PI will study the derivation of effective equations for systems of fermions propagating in a random medium, in the context of the weakly disordered manybody Anderson model with mean field interaction. The analysis of classical Boltzmann equations via Wigner transform and methods of dispersive nonlinear partial differential equations will be further developed. An important outcome of the analysis of these problems is the development of new mathematical techniques merging methods from nonlinear partial differential equations, harmonic analysis, and quantum field theory.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.