报告题目：Two-parameter localization for eigenfunctions of a Schrödinger operator in balls and spherical shells

报告人：张智民

报告时间：6月20日 10:00-11:30

报告地点：五教5106

摘要：

We investigate the two-parameter high-frequency localization for the eigenfunctions of a Schrödinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number l and the principal quantum number k tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [B.-T. Nguyen and D. S. Grebenkov, SIAM J. Appl. Math. 73, 780–803 (2013)]. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the l–k ratio. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Furthermore, we discover a novel phase transition for the eigenfunctions in spherical shells as the l–k ratio crosses a critical value. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. In the critical case, the eigenfunctions are localized around the inner boundary. In the subcritical case, no localization could be observed, giving rise to localization breaking.

报告人简介：

张智民，我校学士（1982）硕士（1985）美国马里兰大学（University of Maryland，College Park）博士（1991）美国韦恩州立大学(Wayne State University) 教授（since 2002），北京计算科学研究中心客座教授，教育部 “CJ学者”讲座教授；现任和曾任10个国内外数学杂志编委，包括Mathematics of Computation、Journal of Scientific Computing、Numerical methods for Partial Differential Equations 、Journal of Computational Mathematics、Communications on Applied Mathematics and Computation、CSIAM Transaction on Applied Mathematics、《数学文化》等；发表SCI论文200余篇。张智民教授长期从事计算方法、尤其是有限元方法的研究，在单元构造、超收敛、后验误差估计等领域的研究取得了多项创新成果，在国际上第一个建立起广为流行的ZZ离散重构格式的数学理论，所提出的多项式保持重构（Polynomial Preserving Recovery—PPR）方法2008年被大型商业软件COMSOL Multiphysics 采用并沿用至今。