Spectral Study of Multidimensional Almost-Periodic Schroedinger Operators

【摘要】karpeschins是豫西的主要地区。Karpeshina之前的研究是关于具有周期势的多维薛定谔算子的微扰理论。我们遇到了一个小分母问题，考虑到拉普拉斯函数在高能区被周期势扰动。这是因为多维周期薛定谔算子的blocheigenen值非常密集地位于高能区。PI开发了一种先进的微扰理论方法来处理这个小分母问题。余。Karpeshina证明了多维周期薛定谔算子在高能区的大多数广义特征函数都接近于无摄动特征函数；对于每一个足够大的能量，薛定谔方程的解都有一个广泛的接近平面波的解集。π将证明，即使在几乎周期的情况下，多维薛定谔算子的许多广义特征函数在高能量区域接近于无摄动的特征函数；对于每一个足够大的能量，薛定谔方程的解都有一个广泛的接近平面波的解集。要克服的主要困难是小分母问题，由于波在非局部偏离规则结构的固体中的传播过程特别复杂，在几乎周期势的情况下比在周期情况下要复杂得多。PI是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，以产生一种适用于近周期势的新技术。具有近周期势的薛定谔算符在物理学中用于描述具有不规则内部结构的固体，例如合金、陶瓷、玻璃、聚合物。对这些算符的光谱研究有助于理解这些材料的导电机制。该项目的目标是了解绝缘体-金属过渡现象。绝缘体-金属过渡意味着，如果一种材料保持在一定温度以下，它就表现为电绝缘体，当温度超过给定材料的某一特性值时，它就突然开始表现为导体。对绝缘体-金属过渡现象的理解对于应用，特别是在电子工业中是极其重要的。

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.