报告题目：Weyl conjecture to fundamental gap conjecture and beyond
报 告 人：Professor Weizhu Bao
所在单位：Department of Mathematics National University of Singapore
报告地点：正新楼209
报告时间：2024年12月27日 10:00-11:00
报告摘要:In this talk, I will begin with the Weyl's law and Weyl conjecture on the asymptotics of eigenvalues of theLaplacian and Schroedinger operators (LO/SO) on bounded domains with Dirichlet boundary condition. Based on our recent numerical results by using a spectral method, I report some information on the reminder in the Weyl conjecture for the LO/SO. In addition, a generalized Weyl's law for the fractional Schrodinger operator (FSO) is proposed. Then I review the fundamental gap conjecture -- difference between the first two smallest eigenvalues -- of the LO/SO. Again, based on our recent asymptotic and numerical results, we propose a gap conjecture on the fundamental gap of the FSO. In addition, different gaps of eigenvalues of the FSO are discussed and the ``unfolding'' gaps statistics of FSO is reported. Finally, fundamental gaps on energy and chemical potential of the Gross-Pitaevskii equation are studied asymptotically and numerically.
报告人简介：Professor Weizhu BAO is Provost's Chair Professor at Department of Mathematics, and Vice Dean for Graduate Studies and Academic Affairs of Faculty of Science, National University of Singapore. He got his PhD from Tsinghua University in 1995. He has made significant contributions in modeling and simulation of Bose-Einstein condensation, solid-state dewetting and geometric flows; and multiscale methods and analysis for highly oscillatory PDEs. He was awarded the Feng Kang Prize in Scientific Computing by the Chinese Computational Mathematics Society in 2013. He was an Invited Speaker at the International Congress of Mathematicians (ICM) in 2014. He was elected a Fellow of the American Mathematical Society (AMS Fellow), a Fellow of the Society of Industrial and Applied Mathematics (SIAM Fellow) and a Fellow of Singapore National Academy of Science (SNAS Fellow) in 2022. In 2024, he was elected as an Officer-at-large of the International Council of Industrialand Applied Mathematicians (ICIAM).
