Spectral and Transport Properties of Multidimensional Almost-Periodic Schroedinger Operators

用具有近周期势的薛定谔方程来描述非周期晶体。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来定义的。传输特性是基于光谱特性的。因此，对薛定谔方程的谱和输运性质的研究导致了对非周期晶体中电导率机制的理解。金属-绝缘体过渡现象在应用中尤为重要。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某个临界值时，材料突然从导电体转变为绝缘体。金属-绝缘体跃迁可以用相应薛定谔方程的谱和输运性质进行数学描述。绝缘体在低能对应于局域本征函数（局域化），而导体在高能对应于非局域本征函数扩展态。导体和绝缘体也对应着不同类型的传输。该项目的目标是描述多维近周期薛定谔算子在高能区域的扩展状态，并研究该区域的弹道输运。弹道输运意味着电子几乎可以自由移动，形成电流。由于缺乏周期性，通常用于研究该算子的“周期性”技术不再适用，必须开发新的技术。PI将开发Kolmogorov-Arnold-Mozer方法的一种新的修正（Multiscale Analysis in Space of Momenta）来解决这个问题。自然界中固体的种类繁多，它们具有不同的物理性质：导电性、导热性、弹性系数等。这种性质的多样性可以用固体的内部结构来解释：首先，由构成固体的原子的类型来解释，其次，非常重要的是，由固体中原子的排列来解释。例如，金刚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。固体物理学中的一个深刻问题是如何解释固体的微观结构与其宏观性质之间的联系。在我们这个时代，随着新工业的发展，能够生产具有规定纳米级或/和原子结构的材料，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，因为它为工业生产更多具有所需性质的材料提供了机会。长期以来，所有被研究的材料都是由周期性的原子阵列组成的，或者是无定形的。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。非周期晶体是一种长范围有序结构，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、矿物质、金属合金和某些蛋白质。事实证明，这些材料具有与晶体和非晶态物质完全不同的性质。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.