Many-body System

第一个项目，量子冷气体，旨在研究相互作用玻色系统在低温下的基本性质。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限制或扩展系统的热力学限制的陷阱系统。此外，我们还建议删除现行的假设上的积极的相互作用潜力。一个与库仑相互作用有关的带电玻色系统基态能量的问题也将被考虑。第二个项目，随机矩阵的本征态和本征值，涉及本征态的能隙分布和局域-离域跃迁。我们重点讨论了带矩阵系综、广义Wigner系综、Wishart系综和辛系综。目标是证明这些系综的局域本征值统计的普适性和带矩阵本征态的定域-离域跃迁。这是该提案的第一个项目，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。第二个项目旨在确定半导体和其他无序系统的导电特性。这些系统的数学模型以最简单的形式由具有随机元素的矩阵给出。我们的项目旨在提供严格的证明，传导确实发生在随机矩阵模型。

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.