Topics in spectral theory of almost periodic operators.

谱理论是研究无穷维算符谱的数学分支。它在物理上的重要性在于量子力学，在量子力学中，像能量这样的物理可观测物是由作用于希尔伯特空间的自伴随算子表示的。量子粒子的运动是用薛定谔方程来描述的，薛定谔方程的特征是一个叫做哈密顿算子的算子。在所有的线性算子中，我们主要研究离散薛定谔算子。它们有一个简单的定义，即一个（离散的）拉普拉斯算子和一个称为势的乘法算子的和，它作用于晶格上的函数，该类足够丰富，可以展示许多谱理论的更一般的现象。我们感兴趣的势属于遍历场，它是由动力系统产生的。无序系统理论中的一个中心问题是算子的谱性质如何依赖于底层动力系统。在研究光谱时，感兴趣的问题包括它的结构和类型。频谱可以进一步分解为绝对连续（AC）、纯点（PP）和奇异连续（SC）部分。当算子描述量子粒子的哈密顿量时，谱型负责粒子的性质（例如，介质是导体还是绝缘体）。最简单的薛定谔算符由周期算符组成，它们对应于有限动力系统。在这种情况下，频谱是纯粹绝对连续的，具有带结构（实线上的间隔集合）。我们计划关注更丰富的一类近周期势，其中最简单的例子是通过沿着无理性旋转轨迹采样一个连续函数获得的：也就是说，我们从圆上的一点开始旋转它的角度是π的无理性倍数，这个无理性常数被称为相位。对几乎周期势的兴趣在于它丰富而奇特的谱性质。作为一个例子，我们有几乎Mathieu算子，在过去的几十年里，由于其朴素的定义（势能是余弦的乘法）和它的谱的丰富性质（谱是康托尔型集合，可以有所有三种（AC， SC和PP）类型的谱）的结合而被广泛研究。几乎周期算符的（SC）和（PP）部分非常敏感地依赖于无理数相的丢芬图性质（无理数与有理数的近似程度）。例如，对于非常近似的不合理相位，几乎Mathieu算子具有SC谱，但是对于一组相位，它具有PP谱，这组相位在测量意义上要大得多。我们项目的目标是探索几乎周期算子的性质，超越一维非理性旋转的研究情况。推广的一个方向是作用于条上的算子。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算子周期逼近的频带长度。根据索利斯的一个似是而非的论点，这应该与最慢的李亚普诺夫指数有关。即使对于一维算子，这也没有得到完全的数学证明；特别是，对于由不合理旋转定义的几乎周期算子，尚不清楚该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它可能应用于谱的度量性质（测度，豪斯多夫维数）的研究；我们计划在项目的后续阶段考虑这些申请。

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.