Spectral Problems of Mathematical Physics Related to Novel Materials Science and Photonics

本研究项目涉及数学物理中出现的算符的谱理论——例如，在量子力学中，适当算符的谱值给出了相关量子态的能级。人们对研究新材料科学的各种光谱理论问题越来越感兴趣。长期以来，这类主题一直被认为与金属和半导体的物理性质有关，由于最近新材料和超材料的发展，例如石墨烯、拓扑绝缘体、碳纳米管和光子晶体等，这些主题得到了新的推动。解决这些问题是本研究的主要目的。该项目将导致新材料科学、凝聚态物理、化学和光子学的关键技术和结果的发展。研究生将参与和培训这一重要的跨学科研究领域。还计划举办一系列关于新材料科学数学的国际研讨会。所涉及的各种相互关联的问题可以分为四个广泛的领域，第一个是关于周期性介质中色散关系的几何。本文将讨论谱隙的存在性和数量、DR的一般行为和极值的位置、DR的不可约性和Dirac点的存在性等问题。所有这些问题都是理解金属和半导体性质的核心，以及诸如二维晶体（石墨烯，石墨烯等），拓扑绝缘体和光子晶体等新材料。克服已知的万尼尔函数基存在的拓扑障碍（对数值计算很重要）的问题也将在这里讨论。第二个要处理的领域涉及阈值效应，它出现在光谱的相关边缘附近。其中特别包括格林函数的精确渐近性、均质化、刘维尔型性质和减慢光的材料设计。第三个领域涉及薄结构，并将解决薄图形或表面结构的建模和特性问题。在模拟光子晶体、光子波导、量子线和其他应用中自然会出现这些问题。第四个领域致力于狄利克雷和诺伊曼特征函数的节点模式（Chladni图形）。

This research project is concerned with spectral theory of operators arising in mathematical physics -- in quantum mechanics, for example, the spectral values of appropriate operators give energy levels of associated quantum states. One witnesses the currently growing interest in studying various spectral theory issues of novel materials science. Such topics, known for a long time to be related to physical properties of metals and semiconductors, received a new boost due to the recent development of novel materials and metamaterials, such as graphene, topological insulators, carbon nanotubes, and photonic crystals, to name a few. Addressing these is the main thrust of this research. This project will lead to the development of techniques and results crucial for novel materials science, condensed matter physics, chemistry, and photonics. Graduate students will be involved and trained in this important interdisciplinary area of research. Running a series of international workshops on mathematics of novel materials science is also planned.The variety of interconnected problems approached can be grouped into four broad areas, the first concerning the Geometry of Dispersion Relations (DR) in periodic media. Here various issues such as existence and number spectral gaps, generic behavior and location of extrema of DR, irreducibility of DR, and existence of Dirac points will be addressed. All these problems are at the heart of understanding properties of metals and semiconductors, as well as novel materials such as 2D crystals (graphene, graphynes, etc.), topological insulators, and photonic crystals. The problem of overcoming the known topological obstacles to the existence of bases of Wannier functions (important for numerical computations) will be also attacked here. A second area to be addressed concerns threshold effects, which arise near and at the relevant edges of the spectrum. These include, in particular, precise asymptotics of Green's functions, homogenization, Liouville type properties, and design of materials slowing down light. The third area concerns thin structures and will address issues of modeling and properties of thin graph-like or surface-like structures. These arise naturally in modeling photonic crystals, photonic waveguides, quantum wires, and other applications. The fourth area is devoted to nodal patterns (Chladni figures) of Dirichlet and Neumann eigenfunctions.