报告人：张智民（韦恩州立大学）

邀请人：李东方

报告时间：2023年8月24日（星期四）9:00-11:00

报告地点：腾讯会议：513 352 376

报告题目：Some mathematical aspects of Anderson localization: boundary effect, multimodality, bifurcation

报告摘要：Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.

报告人简介：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。