具有预定局部结构的离散拟周期集

具有预定覆盖簇的离散拟周期集

Discrete quasiperiodic sets with predefined covering cluster

在四维、五维或六维超空间中，以单位超立方体的投影为选择窗口，用条带投影方法可以得到准晶物理中一些最显著的镶嵌和离散准周期集。我们提出了一些数学结果，允许使用这种非常优雅的方法在超空间的维度高得多，并产生离散的准周期集与一个更复杂的局部结构，从相应的覆盖集群。利用我们的计算机程序，在几分钟内就可以得到这些集合中的数百个点。

Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions

We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schrödinger operators $H = \varepsilonΔ+ V$ with potential $V$ sampled with Diophantine frequency $α$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,α)$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.

Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials

We study discrete quasiperiodic Schrödinger operators on $\ell^2(\zee)$ with potentials defined by $γ$-Hölder functions. We prove a general statement that for $γ>1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.

离散拟周期算子谱测度的连续性

Continuity of the measure of the spectrum for discrete quasiperiodic operators

研究了$l^2(Z)$上的离散Schroedinger算子$(H_{α,θ}ψ)(n)= ψ(n-1)+ψ(n+1)+f(αn+θ)ψ(n)$，其中$f(x)$是周期为1的解析周期函数.在$H_{α,θ}$的李雅普诺夫指数为正的条件下，证明了一个将$H_{α,θ}$的谱测度与其标准有理逼近的谱测度联系起来的一般定理.对于几乎马蒂厄算子（$f(x)=2λ\cos 2πx$），可以得出对于所有实数$θ$、$λ\ne\pm 1$和所有无理数$α$，谱的测度等于$4|1-|λ||$。

生成二十面体三壳层团簇准周期堆积的算法

Algorithm for generating quasiperiodic packings of icosahedral three-shell clusters

条带投影法是产生具有预定局部结构的准周期图案的最重要方法。我们已经得到了一个非常有效的算法，这种方法允许使用它在超空间的非常高的维度。在math-ph/0504036中给出了二维簇的算法版本和十次双壳层簇的应用（10维超空间中的条带投影）。在这种情况下使用的FORTRAN 90程序是非常快的（在3分钟内获得700-800点）。我们提出了一个应用程序，我们的算法的三维集群。将三维物理空间嵌入到31维超空间中，利用条带投影方法，生成由正二十面体的12个顶点组成的三壳层二十面体簇的互穿平移副本的准周期堆积第一壳层是正十二面体的20个顶点（第二壳层），第二壳层是二十半十二面体的30个顶点（第三壳层）。在个人计算机奔腾4与Fortran PowerStation版本4.0（微软开发工作室），我们获得400-500分，在10分钟内。更多详细信息、参考书目和样本可以在网站上找到：http://fpcm5.fizica.unibuc.ro/~ncotfas/

Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.