一维准晶IV的均匀光谱性质。拟Sturmian势

Hierarchical structures in Sturmian dynamical systems

Paper withdrawn; will be replaced by revised version containing application to lattice models as well. We study hierarchical properties of Sturmian words. These properties are similar to those of substitution dynamical systems. This approach allows one to carry over to Sturmian dynamical systems methods developed in the context of substitutions. For example, it allows for a very simple proof of a uniform ergodic type theorem for additive functions taking values in a Banach space. We then focus on establishing various versions of uniform subadditive ergodic type theorems. The main result states…

准晶研究中出现的薛定谔算子的谱性质

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

本文综述了与准晶数学模型相关的自伴算子（尤其是薛定谔算子）的谱性质，重点介绍了一维情形下的关键例子（如斐波那契哈密顿量）及近期引入的双曲动力学方法，并通过数值计算展示已知严格结果并提出猜想。

Fourier quasicrystals and discreteness of the diffraction spectrum

We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.

一维准晶Lebesgue测度零的奇异谱

Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals

证明了严格遍历动力系统的一维离散薛定谔算子的谱与李雅普诺夫指数的零点集一致的充要条件是李雅普诺夫指数一致存在.这是用来获得的Cantor谱的零Lebesgue测度的所有非周期子移位具有一致的正权重。这涵盖了，特别是，所有的非周期子移位所产生的原始替代包括新的例子，例如鲁丁-夏皮罗替代。我们的调查不是基于痕迹地图。相反，它依赖于A. Furman和作者的一个一致遍历定理。

一维准晶谱理论中的Gordon型论点

Gordon-type arguments in the spectral theory of one-dimensional quasicrystals

本文综述了一维离散薛定谔算子的谱理论的最新进展，其中薛定谔算子的势是由代换和圆映射产生的。我们讨论如何发生的局部重复结构允许广义本征函数的估计。在最近的应用程序中，这种一般的方法是几乎肯定和统一的结果，没有本征值以及连续性的谱措施方面的Hausdorff措施。

Quasicrystals with discrete support and spectrum

We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.

Fourier quasicrystals and distributions on Euclidean spaces with spectrum of bounded density

We consider temperate distributions on Euclidean spaces with uniformly discrete support and locally finite spectrum. We find conditions on coefficients of distributions under which they are finite sum of derivatives of generalized lattice Dirac combs. These theorems are derived from properties of families of discretely supported measures and almost periodic distributions.

准晶与概周期性

Quasicrystals and almost periodicity

本文在欧氏空间的一致离散子集空间$U$上引入拓扑${\cal T}$。假设$U$中的$S$允许唯一的自相关度量。$S$的衍射测度是纯原子的当且仅当$S$在$(U,{\cal T})$中是概周期的。这个结果将理想化的准晶与几乎周期性联系起来。在遍历点过程的上下文中，自相关测度是已知存在的。然后，衍射测度是纯原子的当且仅当动力系统具有纯点谱。作为一个例子，我们研究变形模型集。

Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. In this context, we characterize the spectrum of these operators by non-uniformity of the transfer matrices and the set where the Lyapunov exponent vanishes. Adapting this result to subshifts satisfying the so-called Boshernitzan condition, it turns out that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated b…

On the spectrum of 1D quantum Ising quasicrystal

We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan-Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is cont…

Purely singular continuous spectrum for CMV operators generated by subshifts

We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators corresponding to a class of Sturmian subshifts. Lastly, we prove almost sure absence of point spectrum for CMV operators corresponding to some subshifts generated by a coding of a rotation.

具有周期奇异位势的Schrödinger算子

Schrödinger Operators with Periodic Singular Potentials

我们证明了具有来自空间W^{-1}_{2，unif}（R）的奇异势的形式Schr dinger算子可以自然地定义为自伴的和有界的下算子，它们在一致预解式意义下连续地依赖于W^{-1}_{2，unif}（R）-范数下的势.在周期奇异势的情况下，我们也建立了纯绝对连续性和带隙结构的频谱，从而推广了一些经典的结果奇异势的一维准晶理论。