Spectral and Transport Properties of Multidimensional Almost-Periodic Schroedinger Operators
多维准周期薛定谔算子的谱和输运性质
基本信息
- 批准号:1201048
- 负责人:
- 金额:$ 13.69万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-09-01 至 2017-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
本文用具有准周期势的薛定谔方程描述无周期晶体。电子在这种晶体中的运动由所谓的薛定谔方程的输运性质定义。输运性质基于光谱性质。 因此,薛定谔方程的光谱和输运性质的研究导致了解非周期晶体中的导电机制。 金属-绝缘体转变的现象对于应用特别重要。金属-绝缘体转变意味着在接近零度的温度下,当控制固体内部电子能量的外部参数超过某个临界值时,材料突然从电导体转变为绝缘体。金属-绝缘体转变可以用相应的薛定谔方程的光谱和输运性质进行数学描述。绝缘体在低能量下对应于局部本征函数(局部化),而导体在较高能量下对应于非局部本征函数扩展态。 导体和绝缘体也对应于不同类型的输运。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,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,通常的“周期性”技术的研究这个运营商不再工作,新的技术必须开发。PI将开发一种新的修改(动量空间中的多尺度分析)来解决这个问题。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、热导率、弹性系数等。这些性质可以用固体的内部结构来解释:第一,由构成固体的原子的类型决定,第二,非常重要的,由固体中原子的排列决定。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解释固体的微观结构与宏观性质之间的联系。在我们的日子里,随着新产业的发展,能够生产具有规定的纳米级或/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,因为它为工业提供了生产更多具有所需性质的材料的机会。很长一段时间以来,所有被研究的材料都是由原子的周期性排列组成的,或者是无定形的。然而,近几十年来,人们发现了一类新的固态物质,称为非周期晶体,它是一种长程有序结构,但没有严格的晶格周期性。它存在于各种材料中:有机和无机化合物,矿物,金属合金和一些蛋白质。事实证明,这种材料具有与晶体和非晶体物质完全不同的性质。它们具有巨大的应用潜力。
项目成果
期刊论文数量(0)
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会议论文数量(0)
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Ioulia Karpechina其他文献
SPECTRAL PROPERTIES OF DISPLACEMENT MODELS by STEVEN BAKER GUNTER STOLZ, COMMITTEE CHAIR
位移模型的光谱特性 作者:STEVEN BAKER GUNTER STOLZ,委员会主席
- DOI:
- 发表时间:
2007 - 期刊:
- 影响因子:0
- 作者:
R. Brown;Ioulia Karpechina;R. Kawai;B. Kunin;G. Stolz - 通讯作者:
G. Stolz
Ioulia Karpechina的其他文献
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{{ truncateString('Ioulia Karpechina', 18)}}的其他基金
Iterative Methods in Analysis of Periodic and Almost Periodic Structures in Quantum Mechanics
量子力学中周期性和准周期性结构分析的迭代方法
- 批准号:
1814664 - 财政年份:2018
- 资助金额:
$ 13.69万 - 项目类别:
Standard Grant
Spectral Properties of Multidimensional Quasi-Periodic Schroedinger Operators
多维准周期薛定谔算子的谱特性
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0800949 - 财政年份:2008
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$ 13.69万 - 项目类别:
Standard Grant
Spectral Study of Multidimensional Almost-Periodic Schroedinger Operators
多维准周期薛定谔算子的谱研究
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0201383 - 财政年份:2002
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$ 13.69万 - 项目类别:
Standard Grant
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9803498 - 财政年份:1998
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