Topics in spectral theory of almost periodic operators.

几乎周期算子的谱论主题。

• 批准号: 2436138
• 金额: --
• 依托单位: Queen Mary University of London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2436138
• 关键词: Topics; spectral; theory; almost; periodic

Spectral theory is the branch of mathematics studying the spectrum of infinite dimensional operators. Its physical importance lies in quantum mechanics, where physical observables such as energy are represented by self-adjoint operators acting on a Hilbert space. The motion of a quantum particle is described by the Schrodinger equation, which features an operator called the Hamiltonian.Among all linear operators, we focus on the class of discrete Schrodinger operators. These have a simple definition as the sum of a (discrete) Laplacian and a multiplication operator called the potential which act on functions on the lattice, the class is rich enough to exhibit many of the more general phenomena of spectral theory. The potentials we are interested in belong to the class of ergodic fields, which are generated using a dynamical system. A central question in the theory of disordered systems is how do the spectral properties of the operator depend on the underlying dynamical system.When studying the spectrum, questions of interest include its structure and its type. The spectrum can be decomposed further into absolutely continuous (AC), pure point (PP) and singular continuous (SC) parts. When the operator describes the Hamiltonian of a quantum particle, the spectral type is responsible for the properties of the particle (for example, whether the medium is a conductor or an insulator). The simplest class of Schrodinger operators consists of periodic operators, which correspond to a finite dynamical system. In this case, the spectrum is purely absolutely continuous, with a band structure (a collection of intervals on the real line).We plan to focus on the richer class of almost-periodic potentials, the simplest example of which is obtained by sampling a continuous function along the trajectory of an irrational rotation: namely, we start at a point on the circle and rotate it by an angle which is an irrational multiple of pi, this irrational constant is known as the phase. The interest in almost periodic potentials lies in its rich and exotic spectral properties. As an example, we have the almost Mathieu operator, extensively studied in the last few decades due to a combination of its innocent-looking definition (the potential is a multiplication by cosine) and the rich properties of its spectrum (the spectrum is a Cantor-type set, and there can be all three (AC, SC and PP) types of spectra). The (SC) and (PP) parts of the spectrum of almost periodic operators depend very sensitively on the Diophantine properties of the irrational phase (how well the irrational number is approximated by rational numbers). For example, the almost Mathieu operator has (SC) spectrum for very well approximated irrational phases but it has a (PP) spectrum for a set of phases which is much larger in the sense of measure.The goal of our project is to explore the properties of almost-periodic operators beyond the well-studied case of one-dimensional irrational rotation. One of the directions of generalisation is to operators acting in a strip. Similarly to the case of one-dimensional operators, the important tool of transfer matrices is available, however, the structure is much richer, as there are several Lyapunov exponents. One of the questions that we plan to address on the first stage of the project is the length of the bands of periodic approximations of the operator. According to a plausible argument of Thouless, this should be connected to the slowest Lyapunov exponent. This has not been fully mathematically proved even for one-dimensional operators; in particular, for almost periodic operators defined by an irrational rotation, it is not clear whether this property is sensitive to the Diophantine properties of the angle. The importance of this question lies in the possible application to the study of metric properties of the spectrum (measure, Hausdorff dimension); we plan to consider such applications on further stages of the project.

光谱理论是研究无限尺寸算子光谱的数学分支。它的物理重要性在于量子力学，在量子力学上，物理可观察的能量（例如能量）由作用于希尔伯特（Hilbert）空间的自动伴侣算子代表。 Schrodinger方程描述了量子粒子的运动，该方程的特征是一个名为Hamiltonian的操作员。在所有线性运算符中，我们专注于离散的Schrodinger操作员的类别。这些具有简单的定义作为（离散的）拉普拉斯式和称为电势的乘法运算符的总和，该算法对晶格的功能作用，该类足够丰富，足以表现出许多更通用的光谱理论现象。我们感兴趣的电位属于使用动力学系统生成的厄基奇字段类别。无序系统理论中的一个核心问题是操作员的光谱特性如何取决于基本的动力学系统。研究光谱时，感兴趣的问题包括其结构及其类型。频谱可以进一步分解为绝对连续的（AC），纯点（PP）和奇异连续（SC）部分。当操作员描述量子粒子的哈密顿量时，光谱类型负责粒子的性质（例如，介质是导体还是绝缘子）。最简单的Schrodinger操作员由定期操作员组成，该操作员与有限的动态系统相对应。 In this case, the spectrum is purely absolutely continuous, with a band structure (a collection of intervals on the real line).We plan to focus on the richer class of almost-periodic potentials, the simplest example of which is obtained by sampling a continuous function along the trajectory of an irrational rotation: namely, we start at a point on the circle and rotate it by an angle which is an irrational multiple of pi, this irrational constant is known as the phase.对几乎周期性电势的兴趣在于其丰富而外来的光谱特性。例如，由于其看起来无辜的定义（潜力是余弦的乘法）和频谱的丰富属性（频谱是Cantor-Type集合，并且可以有三种（AC，SC和PP）类型的频谱），我们将在过去几十年中进行了几乎Mathieu操作员，在过去的几十年中进行了广泛的研究。几乎周期性运算符的光谱的（SC）和（PP）部分非常敏感地取决于非理性阶段的二磷特性（非理性数量的近似值是有理数）。例如，几乎MATHIEU操作员具有（SC）的频谱，用于非常近似的非理性阶段，但它具有（PP）频谱的一组阶段，在衡量标准方面，该阶段的目标是较大的。我们项目的目的是探索几乎是过时的操作员的性质，而不是良好的不相差旋转。概括的方向之一是在脱衣舞中起作用的操作员。与一维操作员的情况类似，可以使用转移矩阵的重要工具，但是，由于有几种Lyapunov指数，该结构要富裕得多。我们计划在项目的第一阶段解决的问题之一是操作员周期性近似的频段。根据您的合理论证，应将其与最慢的Lyapunov指数有关。即使对于一维操作员，这也没有完全数学上的证明。特别是，对于通过非理性旋转定义的几乎周期性操作员，尚不清楚该特性是否对角度的二磷特性敏感。这个问题的重要性在于可能应用对频谱的度量特性的研究（测量，Hausdorff维度）；我们计划在项目的进一步阶段考虑此类申请。