Spectral Properties of Multidimensional Quasi-Periodic Schroedinger Operators

自然界中有各种各样的固体。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，由形成固体的原子的排列来解释。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，随着能够生产具有规定内部结构的材料的新工业的发展，这个问题变得更加重要，因为了解内部结构和宏观性质之间的基本联系将为工业生产更多具有所需性质的材料提供机会。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，但不具有晶格周期性。它存在于各种各样的材料中：有机和无机化合物、矿物质、金属合金，甚至某些蛋白质。拟周期势的薛定谔方程用于描述一类特殊的非腔晶体：调制晶体。对这些方程的光谱研究有助于理解调制晶体中电导率的机制，特别是金属-绝缘体转变现象。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。该项目的目标是描述多维准周期薛定谔算子在高能区的扩展态。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将开发一种新的KAM （Kolmogorov-Arnold-Mozer）方法来解决这个问题。

There is a huge variety of solids in nature. 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 forming 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 inner structures, this problem becomes more important, since understanding fundamental connections between inner structures and macro properties will give opportunities for industry 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 lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals, metallic alloys, even some proteins.The Schroedinger equations with quasi-periodic potentials are used todescribe a particular kind of aperidoc crystals: modulated crystals. Spectral study of these equations leads to understanding of the mechanism of electrical conductivity in modulated crystals, especially, of the phenomenon of the metal-insulator transition. 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 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.The goal of the project is to describe extended states in the high energy region for multidimensional quasi-periodic Schroedinger operators. 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 of KAM (Kolmogorov-Arnold-Mozer) method to solve theproblem.