Asymptotic Analysis of Almost-Periodic Operators of Quantum Mechanics

近三十年来，准周期结构引起了人们越来越多的兴趣。这种兴趣是由于这些介质在固态物理中的重要性。直到20世纪70年代，所有研究的材料都是由周期阵列组成的，或者是无定形的。在过去的几十年里，人们发现了一类新的固态物质，称为非周期晶体。非周期晶体是一种不具有晶格周期性的长程有序结构。它存在于各种各样的材料中：有机和无机化合物、矿物质（包括地壳的很大一部分）、金属合金（在各种压力和温度下），甚至一些蛋白质中。2011年诺贝尔化学奖承认准晶体的发现，准晶体中的原子是长距离有序的，而不是像传统晶体那样周期性地重复排列。本文的研究重点是利用适当的数学模型来研究这类准周期结构的性质。这项研究将导致对调制晶体中电导率机制的理解，特别是对金属-绝缘体转变现象的理解。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。这项研究结合了来自偏微分方程理论、复分析和其他方面的强大仪器。考虑的科目是在纯数学，理论物理和工程之间的接口。所提出的活动涵盖了在物理和工程中有许多应用的近周期结构的一些新老问题。方法和结构相当复杂，引起了数学家和物理学家的极大兴趣。所提出的研究将有助于更好地理解量子力学、流体力学、量子网络理论、光谱理论、光谱几何、光子晶体理论等领域的一些重要问题。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。拟议的努力还包括将研究纳入本科和研究生课程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Quasi-periodic structures have been attracting increasing interest over the last thirty years. This interest is due to the importance of these media in solid-state physics. Until the 1970s all materials studied consisted of periodic arrays or were amorphous. In the last decades a new class of solid-state matter, called aperiodic crystals, has been found. An aperiodic crystal is a long-range ordered structure but without lattice periodicity. It is found in a wide range of materials: organic and inorganic compounds, minerals (including a substantial portion of the earth's crust), metallic alloys (under various pressures and temperatures), and even some proteins. The 2011 Nobel Prize in Chemistry recognizes the discovery of quasicrystals, in which atoms are ordered over long distances but not in the periodically repeating arrangement of traditional crystals. The present research is focused on the investigation of the properties of such quasi-periodic structures using appropriate mathematical models. This study will lead to the understanding of the mechanism of electrical conductivity in modulated crystals, especially, of the phenomenon of the metal-insulator transition.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 the theory of partial differential equations, complex analysis, 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 almost-periodic structures which have a lot of applications in physics and engineering. The methods and constructions are quite intricate and are of great interest to both mathematicians and physicists. The proposed research will lead to a better understanding of some very important questions in quantum mechanics, hydrodynamics, the theory of quantum networks, spectral theory, spectral geometry, the theory of photonic crystals, and many others. The prospective results can explain or/and predict some effects which appear in experiments. Obtained improvements of different methods can be applied to the investigation of other mathematical and physical problems. The proposed effort also includes integrating the research into the undergraduate and graduate curricula.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.