Spectral problems of mathematical physics and material science

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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.