Many-body System

多体系统

• 批准号: 1204086
• 负责人: Jun Yin
• 金额: $ 3万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1204086&HistoricalAwards=false
• 关键词: Many; body; System

The first project, Quantum Cold Gases, aims to study the basic properties of interacting bose systems in low temperature. The main objective is the identification of the second order term of the ground state energies and the free energies of these systems. The quantum systems considered in this project include trapped systems in the Gross-Pitaevskii limit or extended systems in the thermo-dynamical limits. Furthermore, we also propose to remove the prevailing assumptions on the positivity of the interaction potentials. A related problem on the ground state energy of charged bose systems with Coulomb interactions will also be considered. The second project, Eigen-states, Eigenvalues of Random Matrices, concerns the eigenvalue gap distributions and the localization-delocalization transition of eigen-states. We focus on the band matrix ensemble, the generalized Wigner ensemble, the Wishart ensemble and the symplectic ensemble. The goal is to prove the universality of the local eigenvalue statistics of these ensembles and the localization-delocalization transition of eigen-states of band matrices.This first project of this proposal is aimed to develop new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and can thus increase the understanding of physical systems. The second project aims 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 project is designed to provide rigorous proof that conduction does occur in the random matrix models.

第一个项目，量子冷气体，旨在研究低温下相互作用玻色系统的基本性质。主要目的是确定这些系统的基态能量和自由能的二阶项。在这个项目中考虑的量子系统包括在Gross-Pitaevskii极限中的捕获系统或在热力学极限中的扩展系统。此外，我们还建议消除对相互作用电位正性的普遍假设。本文还将讨论具有库仑相互作用的带电玻色系统的基态能量问题。第二个课题，特征态，随机矩阵的特征值，研究了特征值间隙分布和特征态的局域-离域转换。重点讨论了带矩阵系综、广义Wigner系综、Wishart系综和辛系综。目的是证明这些系综的局部特征值统计量的普适性和带矩阵的特征态的局域-离域跃迁。本提案的第一个项目旨在开发新的数学工具来处理多体量子系统中的复杂问题。新的数学方法导致不同的观点，从而可以增加对物理系统的理解。第二个项目旨在建立半导体和其他无序系统的导电特性。这些系统的最简单形式的数学模型是由随机条目矩阵给出的。我们的项目旨在提供严格的证据，证明传导确实发生在随机矩阵模型中。