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

这项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。人们见证了目前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。这些话题长期以来一直被认为与金属和半导体的物理性质有关，由于最近石墨烯、拓扑绝缘体、碳纳米管和光子晶体等新材料和超材料的发展，这些话题得到了新的推动。解决这些问题是本研究的主旨。该项目将导致对新材料科学、凝聚态物理、化学和光子学至关重要的技术和成果的发展。研究生将参与并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。所讨论的各种相互关联的问题可归类为四个广泛的领域，第一个领域涉及周期介质中的色散关系几何(DR)。这里将讨论各种问题，例如谱间隙的存在和数、DR极值的一般行为和位置、DR的不可约性和Dirac点的存在。所有这些问题都是理解金属和半导体以及新材料的核心问题，如2D晶体(石墨烯、石墨烯等)、拓扑绝缘体和光子晶体。克服已知的Wannier函数基存在的拓扑障碍(对于数值计算很重要)的问题也将在这里被攻克。第二个需要解决的领域涉及阈值效应，它出现在频谱的相关边缘附近。其中特别包括格林函数的精确渐近性、均匀化、Liouville类型性质，以及减慢光速的材料设计。第三个领域涉及薄结构，将解决薄图状或表面状结构的建模和性质问题。在对光子晶体、光子波导、量子线和其他应用进行建模时，这些问题自然会出现。第四部分是Dirichlet和Neumann特征函数的节点型(Chladni图)。

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.