Aperiodic order: spectral theory, combinatorics, and dynamics

研究的中心对象是势函数为非周期序的Schrodinger算子。在一维空间中，人们对它们的光谱和量子动力学性质的理解有了新的进展，特别是在Fibonacci势和相关模型的情况下，所谓的Sturmian势，它们是一维准晶的标准模型。这是拟议的研究的目标，以扩大该理论的更大类的潜力，在一维和解决高维的情况。一维空间中的一个重要工具是迹映射，它是一个能量指数动力系统，可以用来刻画和研究算子的谱。结合组合配分结果和Gordon型准则，可以得到广义本征函数的好的界，由此可以推导出谱和量子动力学的结果.这似乎是可行的，这种方法是适用于潜在的超越类Sturmian潜力-足够低的复杂性应该足以诱导分区和跟踪地图。在更高的维度上，主要的目标是找到一个类似的Gordon准则，它可以作为组合数学和谱理论之间的联系。非周期序数学是一个年轻的新兴领域，自90年代中期以来，已经有了大量的研究活动。来自谱理论、群论、动力系统、组合学和代数拓扑学等不同学科的研究人员发现了一个共同点，这一共同点是由1984年准晶的发现以及随后对有序和有序结构性质的重新思考所激发的。到目前为止，已经提出了相当多的准分子结构模型。人们正在共同努力研究它们的性质，并阐明为什么准晶存在，它们如何形成，为什么它们是稳定的。关于准晶体的电子输运性质，我们预期准晶体可能会表现出反常的行为。因此，计划从这个角度研究Sturmian和相关模型的输运性质。

The central object of study are Schrodinger operators with potentialsdisplaying aperiodic order. In one dimension there have been recentadvances in the understanding of their spectral and quantum dynamicalproperties, particularly in the case of the Fibonacci potential andrelated models, so-called Sturmian potentials, which are the standardmodels of one-dimensional quasicrystals. It is the goal of the proposedresearch to extend the theory to larger classes of potentials in onedimension and to tackle the higher dimensional case. A crucial tool in onedimension is the trace map, an energy-indexed dynamical system which canbe used to characterize and study the spectrum of the operators. Alongwith combinatorial partition results and Gordon-type criteria one canobtain good bounds on generalized eigenfunctions from which one can deducespectral and quantum dynamical consequences. It appears feasible that thisapproach is applicable to potentials beyond the class of Sturmianpotentials -- sufficiently low complexity should suffice to inducepartitions and trace maps. In higher dimensions the main goal is to findan analog of Gordon's criterion which can serve as a link betweencombinatorics and spectral theory.The mathematics of aperiodic order is a young emerging field that hassparked a lot of research activity since the mid-nineties. Researchersfrom disciplines as diverse as spectral theory, group theory, dynamicalsystems, combinatorics, and algebraic topology have found a commonground that was motivated by the discovery of quasicrystals in 1984and the subsequent reconsideration of the nature of order and orderedstructures. By now, quite a number of structural models for quasicrystalshave been proposed. Joint efforts are being undertaken to investigatetheir properties and shed light on why quasicrystals exist, how they form,why they are stable. Regarding their electronic transport properties, itis expected that quasicrystals may exhibit anomalous behavior. It istherefore planned to study transport properties of Sturmian and relatedmodels from this perspective.