Spectral and Transport Properties of Multidimensional Almost-Periodic Schroedinger Operators

用具有概周期势的薛定谔方程描述非周期晶体。电子在这种晶体中的运动是由薛定谔方程的所谓输运性质定义的。传输特性基于光谱特性。因此，研究薛定谔方程的光谱和输运性质有助于理解非周期晶体的电导机制。金属-绝缘体转变现象在实际应用中尤为重要。金属-绝缘体转变是指在接近零的温度下，当控制固体内部电子能量的外部参数超过一定临界值时，材料的性质突然从导电体转变为绝缘体。金属-绝缘体的转变可以用相应薛定谔方程的光谱和输运性质来数学描述。绝缘体对应于低能下的局域本征函数(局域化)，而导体对应于高能下的非局域本征函数扩展态。导体和绝缘体也与不同类型的运输相对应。该项目的目标是描述多维概周期薛定谔算符在高能区的扩展态，并研究该区域的弹道输运。弹道传输意味着电子几乎可以自由运动，形成电流。由于缺乏周期性，研究这种算子的通常的“周期性”技术不再起作用，必须开发新的技术。PI将在Kolmogorov-Arnold-Mozer方法的基础上发展一种新的修正(Momenta空间多尺度分析)来解决这一问题。自然界中存在着大量的固体，它们具有不同的物理性质：电导率、导热率、弹性系数等。这些性质的变化可以由固体的内部结构来解释：第一，由构成固体的原子的类型来解释，第二，非常重要的，由原子在固体中的排列来解释。例如，钻石和石墨都是由相同的碳原子构成的，它们完全不同的性质是由于原子的不同排列。固体物理学中一个深刻的问题是解释固体的微观结构和宏观性质之间的联系。在我们的时代，随着能够生产具有指定纳米或/和原子结构的材料的新工业的发展，了解内部结构和宏观性质之间的基本联系变得比以往任何时候都更加重要，因为它为工业提供了生产更多具有所需性质的材料的机会。在很长一段时间里，所有研究的材料都是由周期性的原子阵列组成，或者是无定形的。然而，在过去的几十年里，一种新的固态物质被发现，称为非周期晶体。非周期晶体是一种长程有序结构，但没有严格的晶格周期性。它广泛存在于各种材料中：有机和无机化合物、矿物、金属合金和一些蛋白质。事实证明，这种材料的性质与晶体和非晶态物质的性质截然不同。它们具有巨大的应用潜力。

The Schroedinger equation with an almost-periodic potential is used to describe aperidic crystals. Motion of electrons in such crystals is defined by so called transport properties of the Schroedinger equation. Transport properties are based on spectral properties. Thus, the study of the spectral and transport properties of the Schroedinger equation leads to understanding of the mechanism of electrical conductivity in aperiodic crystals. A phenomenon of the metal-insulator transition is particularly important for applications. The metal-insulator transition means that at near zero temperatures a material abruptly changes its properties from an electrical conductor to insulator, when an external parameter, controlling electrons energy inside the solid, passes certain critical value. Metal-insulator transition can be described mathematically in terms of spectral and transport properties of the corresponding Schroedinger equation. The insulator corresponds to localized eigenfunctions (localization) at low energies, while the conductor corresponds to non-localized eigenfunctions extended states at higher energies. Conductors and insulators also correspond to different types of transport. The goal of the project is to describe extended states in the high energy region for multidimensional almost-periodic Schroedinger operators and to investigate ballistic transport in this region. Ballistic transport means that electrons can move almost freely forming an electric current . Because of the lack of periodicity the usual "periodic" techniques for the study of this operator no longer work, and new techniques have to be developed. The PI will develop a new modification (Multiscale Analysis in the Space of Momenta) of Kolmogorov-Arnold-Mozer method to solve the problem.There is a huge variety of solids in nature and they have different physical properties: electrical and heat conductivities, elastic coefficients, etc. This variety of properties can be explained by inner structure of solids: first, by types of atoms constituting a solid, and, second, very important, by the arrangement of atoms in a solid. For example, both diamond and graphite are built from the same atoms of carbon, and their completely different properties are due to different arrangements of atoms. A profound problem in Solid State Physics is to explain the connections between micro structures of solids and their macro properties. In our days, with the development of new industries, which are able to produce materials with prescribed nanoscale or/and atomic structures, understanding fundamental connections between inner structures and macro properties becomes more important than ever, since it gives opportunities for industries to produce more materials with desired properties. For a long time all materials studied consisted of periodic arrays of atoms or were amorphous. However, 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 strict lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals, metallic alloys and some proteins. It turns out such materials have properties which are quite different from those of crystals and amorphous substances. They have a huge potential for applications.