非周期度量图与离散图的谱性质
Spectral properties of aperiodic metric and discrete graphs
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中文总结 AI 辅助
本文研究非周期度量图与离散图的谱性质，证明其谱多为广义康托尔集，建立积分态密度的间隙标记定理，否定干燥十马丁尼问题，并刻画斯特姆梳状图的已实现间隙标签。
中文摘要 AI 辅助
在本论文中，我们研究由动力学定义的非周期度量图与离散图的谱性质。我们的目标是确定：当非周期性通过几何而非势体现时，一维非周期遍历薛定谔算子的谱性质会在多大程度上保留下来。本文所考虑的图灵感来源于一维非周期 tiling，被称为 tiling 图与装饰 $\boldsymbol{\text{Z}}$ 图。对于配备标准拉普拉斯算子的一大类度量 tiling 图，我们证明其谱具有零勒贝格测度，且除了可能存在的离散能量集外，是广义康托尔集。对于装饰 $\boldsymbol{\text{Z}}$ 图，我们进一步证明：对于贝尔纲泛型且勒贝格几乎处处的装饰边长选择，其谱为广义康托尔集。随后，我们研究度量与离散装饰 $\boldsymbol{\text{Z}}$ 图的积分态密度（IDS），证明了一个间隙标记定理，该定理刻画了 IDS 在谱间隙内可取的数值集合；我们还表明，间隙标签包含于生成该图的动力系统对应的施瓦茨曼群中，且相差一个几何缩放因子。最后，我们针对离散斯特姆（Sturmian）装饰 $\boldsymbol{\text{Z}}$ 图研究“干燥十马丁尼问题”，即询问间隙标记定理预测的所有可能值是否确实被 IDS 在谱间隙内达到。我们通过识别因 IDS 的跳跃不连续性而未被达到的大量间隙标签，否定了这一问题；随后证明，远离这些跳跃不连续性时，斯特姆图的周期近似量展现出与标准斯特姆哈密顿量相同的组合结构，并利用这一点获得了斯特姆梳状图已实现间隙标签的显式刻画。
英文摘要
In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schrödinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs.
