Spectral Study of Multidimensional Almost-Periodic Schroedinger Operators

多维准周期薛定谔算子的谱研究

• 批准号: 0201383
• 负责人: Ioulia Karpechina
• 金额: $ 7.78万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-05-01 至 2005-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201383&HistoricalAwards=false
• 关键词: Spectral; Study; Multidimensional; Almost; Periodic

AbstractKarpeschinsThe main area of Yu.Karpeshina's research in the previous years was the perturbation theory for multidimensional Schroedinger operators with periodic potentials. One encounters a small denominator problem, considering the perturbation of the Laplacian by a periodic potential in the high energy region. It comes from the fact that the Blocheigenvalues of a multidimensional periodic Schroedinger operator are located very densely in the high energy region.The PI has developed a method of advanced perturbation theory to treat this small denominator problem. Yu. Karpeshina showed that most of generalized eigenfunctions of the multidimensional periodic Schroedinger operator in the highenergy region are close to the unperturbed ones: for every sufficiently large energy there is an extensive set of solutions of the Schroedinger equation which are close to plane waves.The PI will prove that even in the almost-periodic situation a lot of generalized eigenfunctions of the multidimensional Schroedinger operator are close to unperturbed ones in the high energy region: for every sufficiently large energy there is an extensive set of solutions of the Schroedinger equation which are close to plane waves. The main difficulty to overcome is the small denominator problem, which is much more intricate in the case of almost-periodic potentials then in the periodic case, due to particularly complicated nature of wave propagation processes in solids with non-local deviations from regular structure. The PI suggests an effective approach to the small denominator problem. The methods developed by the PI for the multidimensional periodic Schroedinger operator will be combined with basic ideas of the KAM (Kolmogorov-Arnold-Moser) theory in order to produce a new technique which works for almost-periodic potentials. Schroedinger operators with almost-periodic potentials are used in physics to describe solids with non-regular inner structure, e.g. alloys, ceramics, glasses, polymers. The spectral study of these operators leads to understanding of the mechanism of electrical conductivity in such materials. The goal of the project is to understand the phenomenon of the insulator-metal transition. The insulator-metal transition means that a material behaves as an electrical insulator if it stays below a certain temperature and abruptly starts to act as a conductor when the temperature surpasses a certain value characteristic for a given material. The understanding of the phenomenon of insulator-metal transition is extremely important for applications, particularly in electronics industry.

Karpeshina前几年的主要研究领域是多维Schroedinger算子的微扰理论， 周期电位 一个遇到一个小分母的问题，考虑到扰动的拉普拉斯周期性的潜力在高能量区域。这是由于多维周期Schroedinger算子的Bloch本征值在高能区非常密集，PI发展了一种先进的微扰理论方法来处理这个小分母问题。Yu. Karpeshina证明了多维周期Schroedinger算子在高能区的广义本征函数大多数接近于未微扰的本征函数：对于每一个足够大的能量，薛定谔方程都有一组接近平面波的广泛的解。PI将证明，即使在几乎-在周期情况下，多维Schroedinger算子的许多广义本征函数在高能区接近于未微扰的本征函数：对于每一个足够大的能量，都有一组接近于平面波的薛定谔方程的解。要克服的主要困难是小分母的问题，这是更加复杂的情况下，几乎周期性的潜力，然后在周期性的情况下，由于特别复杂的性质，波的传播过程中的固体与非局部偏离定期结构。 PI提出了解决小分母问题的有效方法。PI为多维周期薛定谔算子开发的方法将与KAM（Kolmogorov-Arnold-Moser）理论的基本思想相结合，以产生一种适用于几乎周期势的新技术。 具有准周期势的薛定谔算符在物理学中用于描述具有非规则内部结构的固体，例如合金、陶瓷、玻璃、聚合物。这些运营商的光谱研究导致在这样的材料中的导电性的机制的理解。该项目的目标是了解绝缘体-金属过渡的现象。 绝缘体-金属转变意味着，如果材料保持低于特定温度，则其表现为电绝缘体，并且当温度超过给定材料的特定值特征时，突然开始充当导体。绝缘体-金属转变现象的理解对于应用，特别是在电子工业中是极其重要的。