Title: Weyl Conjecture to Fundamental Gap Conjecture and Beyond
Speaker: Professor Weizhu Bao
Institution: Department of Mathematics, National University of Singapore
Date & Time: December 27, 2024, 10:00–11:00
Location: Zhengxin Building, Room 209
In this talk, I will start with Weyl's law and the Weyl conjecture concerning the asymptotic behavior of eigenvalues for Laplacian and Schrödinger operators (LO/SO) on bounded domains with Dirichlet boundary conditions. Based on our recent numerical results using a spectral method, I will present insights into the remainder term in the Weyl conjecture for LO/SO. Additionally, a generalized Weyl's law for the fractional Schrödinger operator (FSO) will be proposed.
Next, I will review the fundamental gap conjecture—the difference between the first two smallest eigenvalues—for LO/SO. Drawing on our recent asymptotic and numerical findings, we put forward a gap conjecture for the fundamental gap of FSO. Different eigenvalue gaps of FSO will be discussed, and the "unfolding" gap statistics of FSO will be reported.
Finally, asymptotic and numerical studies on fundamental gaps in energy and chemical potential for the Gross-Pitaevskii equation will be presented.
Professor Weizhu Bao is a Provost's Chair Professor in the Department of Mathematics and Vice Dean for Graduate Studies and Academic Affairs at the Faculty of Science, National University of Singapore. He obtained his PhD from Tsinghua University in 1995.
His significant contributions span modeling and simulation of Bose-Einstein condensation, solid-state dewetting, and geometric flows, as well as multiscale methods and analysis for highly oscillatory partial differential equations (PDEs). In 2013, he received the Feng Kang Prize in Scientific Computing from the Chinese Computational Mathematics Society. In 2014, he was an Invited Speaker at the International Congress of Mathematicians (ICM).
In 2022, he was elected a Fellow of the American Mathematical Society (AMS Fellow), a Fellow of the Society for Industrial and Applied Mathematics (SIAM Fellow), and a Fellow of the Singapore National Academy of Science (SNAS Fellow). In 2024, he was elected an Officer-at-large of the International Council for Industrial and Applied Mathematics (ICIAM).
