学术报告

题      目：From Weyl Conjecture to Fundamental Gap Conjecture and Beyond

报  告  人：包维柱  教授  (邀请人：叶颀 )

                                    新加坡国立大学

时      间：12月15日  16：00-17：00

地     点：数科院西楼111报告厅

报告人简介:

        包维柱教授1995年博士毕业于清华大学，现为新加坡国立大学数学系provost讲席教授，2013年获冯康科学计算奖，2014年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，担任包括SIAM Journal on Scientific Computing等多个国际期刊杂志编委。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。特别是在Bose-Einstein 凝聚的数值方法及应用、高震荡色散类偏微分方程的多尺度算法和分析、无界区域上科学和工程问题的计算等方面取得了多个重要进展。

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.

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