From Weyl conjecture to fundamental gap conjecture and beyond
报告题目：From Weyl conjecture to fundamental gap conjecture and beyond
摘要：In this talk, I will begin with the Weyl's law and Weyl conjecture on the asymptotics of eigenvalues of the Laplacian and Schrodinger 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 a Professor at Department of Mathematics, National University of Singapore (NUS).He got his PhD from Tsinghua Univeristy in 1995 and afterwards he had postdoc and faculty positions at Tsinghua University,Imperial College in UK, Georgia Institute of Technology and University of Wisconsin at Madison in USA. He joined NUS as an Assistant Professor in 2000 and was promoted to Professor in 2009.He had been appointed as the Provost's Chair Professorship at NUS during 2013 -- 2016. His research interests include numerical methods for partial differential equations, scientific computing/numerical analysis,analysis and computation for problems from physics, chemistry,biology and engineering sciences. He has made significant contributions in modeling and simulation of Bose-Einstein condensation, solid-state dewetting and geometric PDEs;and in multiscale methods and analysis for highly oscillatory PDEs.He had been on the Editorial Board of SIAM Journal on Scientific Computing during 2009--2014 and is currently on the Editorial Board of SIAM Journal of Numerical Analysis. He was awarded the Feng Kang Prize in Scientific Computing by the Chinese Computational Mathematics Society in 2013.He has been invited to give plenary and/or invited talks in many international conferences including 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.