Spectral problems of mathematical physics and material science

数学物理和材料科学的光谱问题

基本信息

批准号：
2007408
负责人：
Peter Kuchment
金额：
$ 20.14万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007408&HistoricalAwards=false
关键词：
Spectral problems mathematical physics material

项目摘要

The project is devoted to studying various spectral phenomena arising in mathematical physics and areas of novel material science that have been enjoying increased interest recently, among them photonics, carbon nanostructure, and topological insulators. For such problems, it is important to study the spectral properties of several operators, such as the Schrödinger and Dirac operators, as well as operators on quantum graphs. When dealing with crystalline matter, the operators are usually periodic (possibly perturbed by impurities) with respect to appropriate crystalline groups. Many novel materials and meta-materials, such as graphene, graphynes, carbon nanotubes, topological insulators, photonic crystals, and thin dielectric or electronic structures require such studies, which turn out to be challenging and involve high-level and diverse mathematical tools. The results of the project will significantly benefit the active area of novel materials and meta-materials that carry a high promise of technological revolution, including topological insulators, carbon (and other) nanomaterials, slowing light media, and nano-scale electronics. The results will be disseminated in publications in research journals, research presentations nationwide and internationally, taught in graduate level classes, and addressed in a monograph. The project will address a number of problems in several interconnected areas. These include: dispersion relations and spectra, crucial for understanding the properties of crystalline matter; thresholds effects, including a model of “slow light” media; Morse indices and nodal patterns, aiming at further developing a recent discovery of this connection; thin open book structures, addressing models of branching thin surface-like media; frames of Wannier functions in the presence of topological obstructions (as in topological insulators) and gap absence; and analytic properties of Fermi and Bloch varieties in discrete and continuous cases. The project will lead to further development of mathematical techniques important for novel material science, condensed matter physics, photonics, and chemistry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目致力于研究最近在数学物理学和新型材料科学领域引起的各种光谱现象，这些现象最近一直在增加兴趣，其中包括光子学，碳纳米结构和拓扑绝缘子。对于此类问题，重要的是研究几个操作员的光谱特性，例如Schrödinger和Dirac Operators，以及量子图的操作员。在处理晶体物质时，相对于适当的晶体组，操作员通常是周期性的（可能会受到杂质的扰动）。许多新型的材料和元物质，例如石墨烯，图形，碳纳米管，拓扑绝缘子，光子晶体以及稀薄的饮食或电子结构，都需要进行此类研究，这些研究证明是挑战的，涉及高级和不同的数学工具。该项目的结果将显着使新型材料和元物质的活跃区域受益，这些材料和元物质具有很高的技术革命，包括拓扑绝缘子，碳（和其他）纳米材料，较慢的光培养基和纳米级电子设备。结果将在研究期刊，全国和国际研究的研究期刊的出版物中传播，在研究生级课程中教授，并在专着中解决。该项目将解决多个互连区域的许多问题。其中包括：分散关系和光谱，对于理解晶体物质的特性至关重要；阈值效应，包括“慢光”介质的模型；莫尔斯指数和淋巴结模式，旨在进一步发展这种联系的最新发现；薄的开放书结构，解决分支类似于表面的培养基的模型；在存在拓扑目标（如拓扑绝缘子中）和吸收缝隙的情况下，Wannier的框架功能；在离散和连续情况下，费米和Bloch品种的分析特性。该项目将导致对新型材料科学，凝结物理学，光子学和化学重要的数学技术的进一步发展。该奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响审查标准来评估，被认为是宝贵的支持。