From Weyl conjecture to fundamental gap conjecture and beyond
数学公众报告(120周年校庆系列第40场)
报告题目(Title)：From Weyl conjecture to fundamental gap conjecture and beyond
报告人(Speaker)：包维柱（新加坡国立大学）
地点(Place)：ZOOM ID：83312737137 Password：202209
时间(Time)：2022年9月27日（周二），16:00-17:00
邀请人(Inviter)：蔡勇勇
报告摘要
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.
主讲人简介
包维柱教授现为新加坡国立大学数学系教授，理学院副院长，SIAM Fellow, AMS Fellow，新加坡科学院院士，博士毕业于清华大学，曾获冯康科学计算奖，应邀在韩国举行的第26届国际数学家大会上作45分钟邀请报告。包维柱教授长期从事科学与工程计算研究，主要工作涉及偏微分方程数值方法及其在量子物理、流体和材料中的应用。
