Two-parameter localization for eigenfunctions of a Schrödinger operator in balls and spherical shells

主讲人：张智民

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.

主讲人简介：张智民，美国韦恩州立大学教授，Charles H. Gershenson 杰出学者，曾在世界华人数学家大会45分钟报告。研究方向是偏微分方程数值解，包括有限元，有限体积，谱方法等，发表学术论文200余篇；提出的多项式保持重构Polynomial Preserving Recovery（PPR）格式于2008年被国际上广为流行的大型商业软件 COMSOL Multiphysics 采用，并使用至今。担任或曾任“Mathematics of Computation” “Journal of Scientific Computing” 等9个国际计算数学杂志编委。

邀请人：李东方

时间：2023年5月4日（星期四）20:00-22:00