Spectral properties of periodic differential operators

拟议的活动将导致在数学和理论物理的不同古典和现代领域的研究。本研究结合了偏微分方程理论、复分析、Floquet理论等强大的仪器。考虑的科目是在纯数学，理论物理和工程之间的接口。提出的活动涵盖了周期结构的一些新老问题，这些问题在物理和工程上有很多应用。这些方法和结构相当复杂，数学家和物理学家都对此非常感兴趣。许多理论和应用问题导致周期偏微分方程。例如，量子力学、流体力学、弹性理论、导波理论、均匀化理论、量子网络理论、正散射和逆散射、参数共振理论、光谱理论和光谱几何、光子晶体理论等。提出的研究可能会导致更好地理解这些领域的一些非常重要的问题。总而言之，所提出的活动的智力价值在于发展了几个数学丰富和经典的领域，采用了原始的方法和思想。这些结果可能会在不同的数学领域之间提供新的联系，并最终导致更好地理解一些重要过程的基本物理性质。拟议的研究解决了一些问题，这些问题可能会澄清科学和工程中的许多重要影响和问题。预期结果可以解释或/和预测在实验中出现的一些效应。不同方法得到的改进可以应用于其他数学和物理问题的研究。也许，最有趣的可能应用是量子网络理论，它为未来量子电子设备的快速发展提供了数学基础。

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.