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.
In this project, we studied some basic low temperature properties of dilute quantum gases. When dilute gas of bosons are cooled to temperatures very close to absolute zero, they will become a new state of matter: Boson-Einstein condensate (BEC). Under such conditions, a large fraction of the bosons occupy the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale. These effects are called macroscopic quantum phenomena. The BEC was also explained as the mechanism for superfluid and superconducting materials. Though (BEC) was predicted about 90 years ago, the existence of Boson-Einstein condensate has not been rigorously proved yet. In this project, we rigorously proved the leading and next leading term of the free energy of this type of gases. These quantities are very important for understanding Boson-Einstein condensate. We also studied the local statistics of various random matrices and published more than 15 papers in this field. In the past few decades, random matrix theory (RMT) has become a very active field. One of main reasons is that many questions in other fields can be represented mathematically as matrix problems. Our works on various random matrices confirmed the fundamental believe in this field: local statistics of most random matrix models are determined by symmetries and not the details of the model. Furthermore, our results on covariance matrix provides a better understand on the economic phenomenon and population genetics. Our works on band matrix aims to establish the conducting properties of semi-conductors and other disordered systems.
