Many-Body Quantum Dynamics and Quantum Disorder Systems

第一个项目，量子无序系统和随机矩阵，旨在探索随机矩阵和随机薛定谔方程之间的联系。这些课题的基本问题是本征值间隙分布的Wigner-Dyson普适性和随机薛定谔方程的扩张态猜想。由于扩展态猜想不是目前技术所能达到的，我们建议在两种情况下估计随机薛定谔方程在长时间标度极限下的均方位移：1.随机势是时间相关的，并且具有短时记忆。2.无规势由处于平衡状态的声子场给出。受同样的扩展态猜想的启发，我们建议研究随机矩阵的本征向量的局域化/离域化性质。最后，作为迈向Wigner-Dyson普适性的第一步，我们提出估计微观窗口中随机矩阵的本征值密度与Wigner半圆定律之间的差异。第二个项目，玻色气体动力学，涉及从多体薛定谔方程导出唯象的Gross-Pitaevskii方程。我们还建议精确估计Gross-Pitaevskii方程和多体薛定谔方程之间的误差。第三个项目，轴对称不可压缩Navier-Stokes方程的正则性，旨在证明在某些温和的假设下，轴对称流动的INS的正则性。这项提议的第一个项目旨在确定半导体和其他无序系统的导电性质。这些系统的数学模型以最简单的形式由带有随机项的矩阵给出。我们的第一个项目旨在提供严格的证据，证明在随机矩阵模型中确实存在传导。第二个项目旨在建立格罗斯-皮塔夫斯基方程-控制玻色-爱因斯坦凝聚体(一种最近发现的物质，但近一个世纪前由玻色和爱因斯坦提出理论)动力学的基本方程。该项目旨在为描述这些系统的动力学奠定严密的基础。第三个项目涉及轴对称流动的不可压缩Navier-Stokes方程的正则性。该项目不仅将加深对惯性导航系统方程的理解，而且轴对称流动对于模拟飓风和龙卷风等大气事件也很重要。这项建议中涉及的大多数问题都是在各自的传统领域进行研究的，如随机矩阵、数学物理、凝聚态物理或偏微分方程组。这项建议旨在建立这些学科之间的联系，并培养具有跨学科技术和广阔科学视角的学生和博士后。它还将导致这些不同学科的研究人员之间的合作。

The first project, quantum disordered systems and random matrices, aims to explore the connections between the random matrix and random Schrodinger equations. The fundamental questions in these subjects are the Wigner-Dyson universality for the eigenvalue gap distribution and the extended state conjecture for the random Schrodinger equations. Since the extended state conjecture is out of the reach of the current technique, we propose to estimate the mean square displacement for the random Schrodinger equations in long time scaling limits in two cases: 1. The random potential is time dependent and has a short time memory. 2. The random potential is given by a phonon field in equilibrium. Inspired by the same extended state conjecture, we propose to study the localization/delocalization property of eigenvectors of random matrices. Finally, as the first step toward the Wigner-Dyson universality, we propose to estimate the difference between the density of eigenvalues of random matrices in microscopic windows and the Wigner semicircle law. The second project, dynamics of Bose gas, concerns the derivation of the the phenomenological Gross-Pitaevskii equation from the many-body Schrodinger equations. We also propose to estimate precisely the errors between the Gross-Pitaevskii equation and the many-body Schrodinger equation. The third project, regularity of axisymmetric incompressible Navier-Stokes equations, aims to prove the regularity of the INS for the axisymmetric flow under certain mild assumptions. This first project of this proposal is aimed to establish the conducting properties of semiconductors and other disordered systems. The mathematical model for these systems in the simplest form is given by matrices with random entries. Our first project is designed to provide rigorous proof that conduction does occur in the random matrix models. The second project aim to establish the Gross-Pitaevskii equation---the fundamental equation governing the dynamics of the Bose-Einstein condensate (a recent discovered material but was theorized by Bose and Einstein almost a century ago). This project aims to lay rigorous foundation for the description of the dynamics of these systems. The third project concerns regularity of the incompressible Navier-Stokes equations for axisymmetric flow. This project will not only lead to deeper understanding of the INS equations in general, but the axisymmetric flow is important for modeling atmospheric events such as hurricanes and tornados. Most problems covered in this proposal were studied in their own traditionally fields such as random matrix, mathematical physics, condense matter physics or partial differential equations. This proposal aims to establish connections between these subjects and to train students and postdoctors with interdiscipline technique and broad scientific perspectives. It will also lead to collaborations of researchers in these diverse disciplines.