Spectral properties of quasicrystals via dynamical methods

该项目的重点是发展新方法和改进现代动力系统理论中的现有技术，这些技术可用于研究最突出的准晶体模型的光谱特性，特别是具有斐波那契和斯图尔曼势的离散薛定谔算子。要研究的具体模型是具有Sturmian势的离散薛定谔算子、平方（或立方）斐波那契哈密顿量、斐波那契量子Ising模型、准晶体的量子行走和迷宫模型。我们打算研究这些模型和双曲动力系统之间的深层关系，使我们能够严格证实物理学家先前对准晶体所做的一系列启发式和实验观察。特别是，对于具有Sturmian势的离散薛定谔算符，我们期望表明频谱的Hausdorff维数几乎对每个频率都是相同的。对于平方和三次斐波那契哈密顿量，我们打算提供频谱随耦合常数的变化从区间到康托集的过渡的理解，并在小耦合区中建立态测量密度的绝对连续性。同时，我们希望证明具有大耦合的平方连续体Fibonacci哈密顿量具有混合（交流和奇异连续）类型的状态密度测度，并提供具有此性质的遍历薛定谔算子族的第一个例子。本课题的问题与准晶体的数学模型有关。物理学家和化学家对准晶体的性质进行了大量的研究，在严格的数学分析的基础上对观察到的实验和数值结果进行解释是很重要的。这不仅可以解释现有模型的行为，还可以帮助预测新模型的行为。我们打算开发的工具无疑也会在数学的其他部分找到应用。例如，为了证明某些模型的状态密度测度的绝对连续性，我们研究了奇异测度卷积的绝对连续性问题。这些问题经常出现在数论、分析、动力学、概率论和几何测量理论中。此外，数学不同部分之间的关系（在这种情况下，谱理论和动力系统之间的关系）为数学的统一性提供了漂亮的证据。与项目密切相关的各种问题将被建议给加州大学欧文分校的研究生和本科生，从而引导他们进入或增加他们对科学活动的参与。首席研究员认为这些教育和培训方面是非常必要的。

The project focuses on the development of new methods and the improvement of existing techniques in the modern theory of dynamical systems that can be applied to study spectral properties of the most prominent models of quasicrystals, in particular, of discrete Schrodinger operators with Fibonacci and Sturmian potentials. The specific models to be studied are discrete Schrodinger operators with Sturmian potentials, the square (or cubic) Fibonacci Hamiltonian, the Fibonacci quantum Ising model, quantum walks for quasicrystals, and the labyrinth model. We intend to study the deep relations between these models and hyperbolic dynamical systems that allow us to confirm rigorously a series of previous heuristic and experimental observations regarding quasicrystals made by physicists. In particular, for discrete Schrodinger operators with Sturmian potentials we expect to show that the Hausdorff dimension of the spectrum is the same for almost every frequency. For square and cubic Fibonacci Hamiltonians we intend to provide an understanding of the transition of the spectrum from an interval to a Cantor set with the change of the coupling constant, and establish absolute continuity of the density of states measure in the small coupling regime. Also, we hope to prove that the square continuum Fibonacci Hamiltonian with large couplings has density of states measure of a mixed (a.c. and singular continuous) type, providing the first example of an ergodic family of Schrodinger operators with this property.The problems in this project are related to mathematical models of quasicrystals. The properties of quasicrystals were heavily studied by physicists and chemists, and it is important to give an explanation of the observed experimental and numerical results based on a rigorous mathematical analysis. Potentially this will not only explain the behavior of existing models but will also help to predict the behavior of new ones. The tools that we intend to develop will undoubtedly find applications in other parts of mathematics as well. For example, in order to prove absolute continuity of density of states measures for some models, we are studying the questions on absolute continuity of convolutions of singular measures. These questions appear frequently in number theory, analysis, dynamics, probability, and geometric measure theory. In additin, the relations between different parts of mathematics (in this case, between spectral theory and dynamical systems) give beautiful evidence of the unity of mathematics. Various problems closely related to the project will be suggested to graduate and undergraduate students at UC-Irvine, thereby initiating them into, or increasing their involvement in, scientific activities. The principal investigator considers such educational and training aspects to be very essential.