分数阶和长程安德森模型的泊松特征值统计与动力学去局域化

中文摘要 AI 辅助

我们研究了长程安德森模型（包括分数阶安德森模型）的局部特征值统计和动力学（去）局域化性质。这些系统是在d维格点上，对具有长程（非随机）跳跃项$|T(n,m)| \sim \\\\|n-m\\\\|^{-(d+2\beta)}$（其中$\beta>0$）的算子进行随机安德森型扰动。在足够强的随机势存在下，这些模型几乎必然表现出具有多项式衰减特征向量的稠密纯点谱。我们证明，在强无序区域且$\beta>d/2$时，以确定性谱中任意$E$为中心的局部特征值统计是一个泊松点过程，其强度由态密度函数$n(E)$给出。此外，我们证明在强无序下，这些模型不存在动力学局域化，验证了Disertori等人的猜想。我们通过建立格林函数分数矩的显式尖锐下界来实现这一点，该下界意味着位置算子的大矩是无穷的。因此，这些模型表现出泊松特征值统计和动力学去局域化，即不存在动力学局域化。特别地，这表明在维度$d=1$中，分数阶安德森模型（分数阶拉普拉斯算子$(-\Delta)^\alpha$的随机扰动，其中$\alpha\in(0,1)$）在无序较强且指数$\frac{1}{2}<\alpha<1$时表现出泊松特征值统计和动力学去局域化。这与仅具有最近邻跳跃的通常安德森模型的已知行为形成鲜明对比。

英文摘要

We study local eigenvalue statistics and dynamical (de)-localization properties for long-range Anderson models, including the fractional Anderson model. These systems are random Anderson-type perturbations of operators with long-range (non random) hopping terms of the form $|T(n,m)| \sim \|n-m\|^{-(d+2β)}$ for $β>0$, on the $d$-dimensional lattice. In the presence of a strong enough random potential, these models exhibit dense pure point spectrum with polynomially decaying eigenvectors, almost surely. We show that in this strong disorder regime, and for $β>d/2$, the local eigenvalue statistics centered at any $E$ in the deterministic spectrum is a Poisson point process with intensity given by the density of states function $n(E)$. Moreover, we prove that, at strong disorder, there is no dynamical localization for these models, verifying a conjecture of Disertori et al. We achieve this by establishing explicit, sharp lower bounds on the Green's functions fractional moments which imply that large moments of the position operator are infinite. Hence, these models exhibit Poisson eigenvalue statistics and dynamical delocalization, that is, the absence of dynamical localization. In particular, this shows that in dimension $d=1$, the fractional Anderson model, a random perturbation of the fractional Laplacian $(-Δ)^α$ with $α\in (0,1)$, exhibits Poisson eigenvalue statistics and dynamical delocalization if the disorder is strong and the exponent $\frac{1}{2}<α<1$. This is in stark contrast with what is known for the usual Anderson model that has only nearest-neighbor hopping.

发表机构

• University of Kentucky（肯塔基大学）
• PUC-Rio（天主教里约热内卢大学）
• CY Cergy Paris Université（塞吉-巴黎CY大学）

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