Spectral properties of periodic differential operators

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The proposed activity will lead to research in different classical as well as modern areas of mathematics and theoretical physics. This research combines powerful apparatus from theory of partial differential equations, complex analysis, Floquet theory and others. Considered subjects are at the interfaces between pure mathematics, theoretical physics and engineering. The proposed activity covers some old and new questions for periodic structures which have a lot of applications in physics and engineering. The methods and constructions are quite intricate and are of great interest for both mathematicians and physicists. Many theoretical and applied problems lead to periodic partial differential equations. For instance, quantum mechanics, hydro-dynamics, elasticity theory, the theory of guided waves, homogenization theory, theory of quantum networks, direct and inverse scattering, parametric resonance theory, spectral theory and spectral geometry, theory of photonic crystals and many others. The proposed research may lead to better understanding of some very important questions in these areas. To summarize, the intellectual merit of the proposed activity is in development of several mathematically rich and classical areas employing original approaches and ideas. The results will likely provide new links between different fields of mathematics and eventually lead to better understanding of the fundamental physical nature of some important processes. The proposed research addresses questions that may clarify many effects and problems important in science and engineering. The prospective results can explain or/and predict some effects which appear in experiments. Obtained improvements of different methods can be applied for investigation of other mathematical and physical problems. Perhaps, the most interesting possible application is the theory of quantum networks which provides the mathematical base for quickly developing engineering of future quantum electronic devices.