Geometric Scattering Theory, and Propagation and Decay of Waves

几何散射理论、波的传播和衰变

• 批准号: 1708511
• 负责人: Kiril Datchev
• 金额: $ 18.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708511&HistoricalAwards=false
• 关键词: Geometric; Scattering; Theory; Propagation; Decay

The principal investigator will study mathematical problems in the subject of particle-wave duality. This duality has been successfully investigated for a long time, especially in the setting of closed systems, but many of its important manifestations are poorly understood, especially in the settings of open and mixed systems which are the principal investigator's focus. In open systems the method of complex scaling studied by the principal investigator appears in numerical analysis as the method of perfectly matched layers, where it has been useful in the engineering of nanostructures and microelectromechanical systems. Mixed systems like the ones studied by the principal investigator are important in electromagnetics, communications, optics, quantum waveguides, and semiconductors.The main dynamical object studied by the principal investigator is the trapped set; this is the set of particle trajectories which remain in a bounded set for all time. The geometry of this set is of central importance in determining salient features of wave evolution, such as regularity, decay, and asymptotics. The principal investigator will study these features in the important but still comparatively mysterious situations of heavy trapping, rough coefficients, and noncompact trapped sets.

首席研究员将研究粒子波对偶性的数学问题。这种二元性已经成功地研究了很长一段时间，特别是在封闭系统的设置，但它的许多重要表现形式知之甚少，特别是在开放和混合系统的设置，这是主要研究者的重点。在开放系统中，主要研究者研究的复尺度方法在数值分析中作为完全匹配层的方法出现，在纳米结构和微机电系统的工程中非常有用。主要研究者所研究的混合系统在电磁学、通信、光学、量子波导和半导体中非常重要。主要研究者所研究的主要动力学对象是被困集;这是粒子轨迹的集合，它们始终保持在一个有界集合中。这个集合的几何形状在确定波演化的显著特征，如规律性、衰减性和渐近性方面至关重要。主要研究人员将研究这些功能的重要，但仍然比较神秘的情况下，沉重的陷阱，粗糙的系数，和非紧凑的陷阱集。

项目成果

Wave Asymptotics for Waveguides and Manifolds with Infinite Cylindrical Ends

具有无限圆柱端的波导和流形的波渐近

• DOI: 10.1093/imrn/rnab254
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Christiansen, T J;Datchev, K
• 通讯作者: Datchev, K

Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides

星形波导的分辨率估计、波衰减和无共振区域

• DOI: 10.4310/mrl.2022.v29.n1.a4
• 发表时间: 2022
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Christiansen, T. J.;Datchev, K.
• 通讯作者: Datchev, K.

On the interaction of metric trapping and a boundary

关于度量陷阱和边界的相互作用

• DOI: 10.1090/proc/15460
• 发表时间: 2021
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Datchev, Kiril;Metcalfe, Jason;Shapiro, Jacob;Tohaneanu, Mihai
• 通讯作者: Tohaneanu, Mihai

Seven useful questions in density functional theory

• DOI: 10.1007/s11005-023-01665-z
• 发表时间: 2023-04-09
• 期刊: LETTERS IN MATHEMATICAL PHYSICS
• 影响因子: 1.2
• 作者: Crisostomo, Steven;Pederson, Ryan;Burke, Kieron
• 通讯作者: Burke, Kieron

Semiclassical resolvent bounds for compactly supported radial potentials

紧支撑径向势的半经典解析界限

• DOI: 10.1016/j.jfa.2022.109835
• 发表时间: 2023
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Datchev, Kiril;Galkowski, Jeffrey;Shapiro, Jacob
• 通讯作者: Shapiro, Jacob