Spectral problems of mathematical physics and material science

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。对于这类问题，研究一些算符的谱性质是很重要的，例如薛定谔算符、狄拉克算符以及量子图上的算符。在处理结晶物质时，运算符相对于适当的结晶基团通常是周期性的(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、光子晶体和薄的介电或电子结构，都需要这样的研究，这被证明是具有挑战性的，涉及到高水平和多样化的数学工具。该项目的成果将大大有利于新材料和超材料的活跃领域，这些材料和超材料具有很高的技术革命前景，包括拓扑绝缘体、碳(和其他)纳米材料、慢光介质和纳米电子产品。研究结果将在国内和国际的研究期刊和研究报告中发表，在研究生级别的课堂上讲授，并以专著形式发表。该项目将解决几个相互关联领域的一些问题。这些包括：色散关系和光谱，对于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；Morse指数和节点图，旨在进一步发展最近发现的这种联系；薄开卷结构，寻址薄表面状介质的分支模型；在存在拓扑障碍(如拓扑绝缘体)和没有间隙的情况下的Wannier函数框架；以及离散和连续情况下费米和布洛赫变种的解析性质。该项目将导致对新材料科学、凝聚态物理、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的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.