Iterative Methods in Analysis of Periodic and Almost Periodic Structures in Quantum Mechanics

该项目致力于加深对一类重要固体和具有广泛工程和物理应用的新物质状态（玻色爱因斯坦凝聚，BEC）的理论理解。长期以来，固体物理学研究的大多数材料要么由周期性原子阵列组成，要么是非晶态（玻璃）。然而，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（或准晶体）。非周期性晶体是一种长程有序结构，但没有严格的晶格周期性。它存在于多种材料中：有机和无机化合物、矿物质、金属合金甚至一些蛋白质。到目前为止，准晶体已在手术器械、LED 灯等设计中得到应用，而且这个应用范围还在不断增加。对于周期性和准周期性材料，该项目的目标是解决固体物理学中的基本问题，即解释固体的微观（量子力学）结构与其宏观性质之间的联系。该项目的第三个研究方向是对 BEC 行为的理论研究（Bose 和 Einstein 于 1924 年首次从理论上预测，但直到 1995 年才进行实验）。虽然量子现象在非常小的微观尺度上表现出来，但在大的宏观尺度上，经典牛顿力学可以很好地描述自然，因为 BEC 宏观量子现象变得显而易见。 BEC 有很多潜在的应用，包括量子信息和量子计算机。几名研究生将参与这项研究。 在对 2 维及更高维度的准周期薛定谔算子主题进行严格研究的领域，该项目的目标包括证明高能扩展量子态的存在，研究与此类算子相关的量子输运，以及分析高能稳态 Gross-Pitaevskii 方程解的性质。所使用的核心分析方法是动量空间中的多尺度分析。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。

This project is devoted to furthering the theoretical understanding of an important class of solids and of the new state of matter (Bose-Einstein Condensate, BEC) that have a wide range of engineering and physical applications. For a long time, most of materials studied by Solid State Physics consisted either of periodic arrays of atoms or were amorphous (glasses). However, in the last decades a new class of solid state matter, called aperiodic crystals (or quasicrystals), has been found. An aperiodic crystal is a long range ordered structure, but without strict lattice periodicity. It is found in a wide range of materials: organic and inorganic compounds, minerals, metallic alloys and even some proteins. So far, quasicrystals found applications in the design of surgical instruments, LED lights, etc., and the list is growing. For the periodic and almost periodic materials, the goal of the project is to contribute to the fundamental problem in Solid State Physics of explaining the connections between micro (quantum mechanical) structures of solids and their macro properties. The third line of research in this project is aimed at a theoretical study of behavior of BEC (that was first predicted theoretically by Bose and Einstein in 1924, but was not produced experimentally until 1995). While quantum phenomena are exhibited on very small micro-scales, and on large macro-scales nature is well described by classical, Newtonian mechanics, for a BEC macroscopic quantum phenomena become apparent. There is a long list of potential applications of BEC, including quantum information and quantum computer. Several graduate students will be involved in this research.In the area of rigorous study of topics in quasi-periodic Schroedinger operators in dimension 2 and higher, the goals of the project include proof of existence of extended quantum states at high energies, investigation of quantum transport associated with such operators, and also analysis of properties of solutions to the stationary Gross-Pitaevskii equation at high energies. The central analytical method to be used is a multiscale analysis in the momentum space.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.