Asymptotic Analysis of Almost-Periodic Operators of Quantum Mechanics

量子力学准周期算子的渐近分析

• 批准号: 2306327
• 负责人: Roman Shterenberg
• 金额: $ 22万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306327&HistoricalAwards=false
• 关键词: Asymptotic; Analysis; Almost; Periodic; Operators

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.

在过去的三十年里，准周期结构引起了越来越多的兴趣。这种兴趣是由于这些介质在固态物理学中的重要性。直到 20 世纪 70 年代，所有研究的材料都是由周期性阵列组成或者是非晶态的。在过去的几十年里，人们发现了一类新的固态物质，称为非周期性晶体。非周期性晶体是一种长程有序结构，但不具有晶格周期性。它存在于多种材料中：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（在不同压力和温度下），甚至一些蛋白质。 2011 年诺贝尔化学奖表彰了准晶体的发现，准晶体中的原子在长距离内有序排列，但与传统晶体的周期性重复排列不同。目前的研究重点是使用适当的数学模型研究这种准周期结构的特性。这项研究将有助于理解调制晶体的导电机制，特别是金属-绝缘体转变现象。所提出的活动将导致数学和理论物理的不同经典和现代领域的研究。这项研究结合了偏微分方程、复分析等理论的强大工具。所考虑的学科是纯数学、理论物理和工程学之间的交叉点。拟议的活动涵盖了近周期结构的一些新旧问题，这些问题在物理和工程中有很多应用。其方法和结构相当复杂，并且引起了数学家和物理学家的极大兴趣。拟议的研究将有助于更好地理解量子力学、流体动力学、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非常重要的问题。前瞻性结果可以解释或/和预测实验中出现的一些效应。不同方法所获得的改进可以应用于其他数学和物理问题的研究。拟议的工作还包括将研究纳入本科生和研究生课程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。