通过伯努利随机化实现 Delone 算子的局域化
Localisation for Delone operators via Bernoulli randomisation
AI总结：
本文通过向 Delone 集添加随机点并引入伯努利随机耦合常数，构造可适用多尺度分析的非遍历 Anderson 模型，利用定量唯一延拓原理克服非周期背景势困难，证明具有谱底端动力学局域化的 Delone 集在 Delone 集空间中稠密。
摘要：
Delone 算子是定义在多维欧几里得空间中的薛定谔算子，其势能由给定“单点势”在所有 Delone 集点处平移的总和给出。本文利用随机化方法研究 Delone 算子族的动力学局域化。具体做法是：向 Delone 集适当添加更多点，并在这些附加点上引入独立同分布的伯努利随机变量作为耦合常数。由此得到的具有伯努利无序的非遍历连续 Anderson 模型可适用于最新版本的多尺度分析。本文的新颖之处在于初始长度尺度估计，其证明因非周期背景势的存在而受阻。该估计通过定量唯一延拓原理获得。作为应用，我们得到了关于动力学局域化的概率性和拓扑性结论。其中，我们证明了：若某 Delone 集所对应的 Delone 算子在谱底端表现出动力学局域化，则这类 Delone 集在 Delone 集空间中稠密。
英文摘要：
Delone operators are Schrödinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given "single-site potential" centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
