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

该项目致力于促进对一类重要的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。很长一段时间以来，固体物理学研究的大多数材料都是由原子的周期性排列组成的，或者是无定形的（玻璃）。然而，近几十年来，人们发现了一类新的固态物质--非周期晶体（或准晶体），它是一种长程有序结构，但没有严格的晶格周期性。它存在于各种材料中：有机和无机化合物，矿物，金属合金，甚至一些蛋白质。到目前为止，准晶在手术器械、LED灯等的设计中得到了应用，名单还在增长对于周期性和几乎周期性的材料，该项目的目标是为解释固体的微观（量子力学）结构与其宏观性质之间的联系的固态物理学中的基本问题做出贡献。该项目的第三条研究路线旨在对BEC的行为进行理论研究（BEC在1924年首次由玻色和爱因斯坦在理论上预测，但直到1995年才在实验上产生）。虽然量子现象表现在非常小的微观尺度上，而在大的宏观尺度上，自然是很好地描述了经典，牛顿力学，为BEC宏观量子现象变得明显。BEC有一长串潜在的应用，包括量子信息和量子计算机。本研究将由多名研究生参与。在2维以上的准周期薛定谔算符的严格研究领域，本研究的目标包括证明高能扩展量子态的存在性，研究与这种算符相关的量子输运，以及分析高能稳态Gross-Pitaevskii方程的解的性质。该奖项反映了NSF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。

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.