北航数学论坛学术报告

From Weyl conjecture to fundamental gap conjecture and beyond

包维柱

（新加坡国立大学）

报告时间：10：30-11：30，2024-04-23（星期二）

报告地点： 沙河E-404

内容简介：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：包维柱，新加坡国立大学数学系教授，理学院副院长，新加坡科学院院士、SIAM Fellow（美国工业与应用数学学会院士）、 AMS Fellow（美国数学会院士），本硕博均毕业于清华大学，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。担任SIAM Journal of Numerical Analysis等权威期刊编委。

邀请人：韩德仁