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