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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.

该项目致力于研究数学物理和新材料科学领域中出现的各种光谱现象，这些领域最近受到越来越多的关注，其中包括光子学，碳纳米结构和拓扑绝缘体。对于这样的问题，重要的是研究几个算子的谱性质，例如薛定谔和狄拉克算子，以及量子图上的算子。当处理晶体物质时，算子通常是周期性的（可能受到杂质的干扰）。许多新材料和超材料，如石墨烯，石墨烯，碳纳米管，拓扑绝缘体，光子晶体和薄电介质或电子结构需要这样的研究，这是具有挑战性的，涉及高层次和多样化的数学工具。该项目的结果将显着有利于具有很高技术革命前景的新型材料和超材料的活跃领域，包括拓扑绝缘体、碳（和其他）纳米材料、慢光介质和纳米级电子产品。研究结果将在研究期刊出版物中传播，在全国和国际上进行研究演示，在研究生课程中授课，并在专著中讨论。该项目将解决若干相互关联领域的若干问题。其中包括：色散关系和光谱，对理解晶体物质的性质至关重要;阈值效应，包括“慢光”介质的模型;莫尔斯指数和节点模式，旨在进一步发展最近发现的这种联系;薄开卷结构，解决分支薄表面状介质的模型;框架的Wannier功能存在的拓扑障碍物（如拓扑绝缘体）和差距的情况下，和分析性质的费米和布洛赫品种在离散和连续的情况下。该项目将导致新材料科学、凝聚态物理学、光子学和化学的重要数学技术的进一步发展。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。