学术报告20230817：Two-parameter localization for eigenfunctions of a Schrodinger operator in balls and spherical shells

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

报告人：张智民 教授 (美国韦恩州立大学)

摘 要：Here, 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，中国科学技术大学学士（1982）硕士（1985）马里兰大学（University of Maryland，College Park）博士（1991）韦恩州立大学(Wayne State University)教授（2002-）北京计算科学研究中心讲座教授（2012-2022）教育部“长江学者”（2010）国家引进海外高层次人才（2012），现任和曾任10个国内外数学杂志编委，包括Mathematics of Computation（2009-2017）、Journal of Scientific Computing（2011-2017）、Numerical methods for Partial Differential Equations（2013-）、Journal of Computational Mathematics（2007-）、等, 在计算数学顶级刊物SIAM Journal on Numerical Analysis、Mathematics of Computation、Numerische Mathematik等杂志上发表SCI等学术论文200余篇。2014年以前，主持过10个美国国家基金会的项目；2014年以后主持过9个国家自然科学基金委员会的重点、面上、天元、国际交流等项目。

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、后验误差估计和自适应算法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。

报告时间：2023年8月17日，下午15:00-16:30

报告地点：明远楼410

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