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.
