Topics in spectral theory of almost periodic operators.

谱理论是研究无穷维算子谱的数学分支。它的物理重要性在于量子力学，其中物理可观测量，如能量，由作用于希尔伯特空间的自伴算子表示。量子粒子的运动由薛定谔方程描述，其中包含一个称为哈密顿算子的算子。在所有线性算子中，我们重点讨论离散薛定谔算子。这些都有一个简单的定义作为总和的（离散）拉普拉斯算子和乘法算子称为潜在的作用于功能上的晶格，类是丰富的，足以展示许多更普遍的现象谱理论。我们感兴趣的潜力属于遍历领域，这是由动力系统产生的类。无序系统理论的一个中心问题是算符的谱性质如何依赖于底层动力系统。当研究谱时，感兴趣的问题包括它的结构和类型。该谱可进一步分解为绝对连续（AC）、纯点（PP）和奇异连续（SC）部分。当算符描述量子粒子的哈密顿量时，谱类型负责粒子的性质（例如，介质是导体还是绝缘体）。最简单的薛定谔算子类由周期算子组成，它们对应于有限动力系统。在这种情况下，光谱是纯粹绝对连续的，具有带结构（真实的线上的区间集合）。我们计划关注更丰富的几乎周期势类，其中最简单的例子是通过对沿着无理旋转的轨迹的连续函数进行采样而获得的：也就是说，我们从圆上的一点开始，并将其旋转一个角度，该角度是π的无理倍数，这个无理常数称为相位。概周期势的有趣之处在于它丰富而奇异的谱性质。作为一个例子，我们有几乎马蒂厄算子，在过去的几十年里，由于其看似无辜的定义（势是余弦的乘法）和其谱的丰富性质（谱是康托型集，并且可以有所有三种（AC，SC和PP）类型的谱）的组合，它被广泛研究。概周期算子谱的（SC）和（PP）部分非常敏感地依赖于无理相位的丢番图性质（无理数如何被有理数近似）。例如，几乎Mathieu运营商有（SC）的频谱非常近似的无理阶段，但它有一个（PP）的频谱的一组阶段，这是更大的意义上measure.The我们的项目的目标是探索的性质几乎周期运营商超出了良好的研究情况下，一维无理旋转。推广的方向之一是算子作用在一个带。类似于一维算子的情况，传递矩阵的重要工具是可用的，然而，结构要丰富得多，因为有几个李雅普诺夫指数。我们计划在项目的第一阶段解决的问题之一是运营商的周期近似带的长度。根据一个看似合理的论点，这应该与最慢的李雅普诺夫指数有关。即使对于一维算子，这一点也没有得到充分的数学证明;特别是对于由无理旋转定义的概周期算子，不清楚这个性质是否对角度的丢番图性质敏感。这个问题的重要性在于可能的应用研究的度量性质的频谱（措施，豪斯多夫维数），我们计划考虑这样的应用程序的进一步阶段的项目。

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.