在嵌套分形上带有泊松随机环境的从属布朗运动的IDS
IDS for subordinate Brownian motions in Poisson random environment on nested fractals
AI总结：
本文研究了在嵌套分形上带有泊松随机环境的从属布朗运动的IDS奇异性，通过将随机势能转化为合金型势能，拓展了伯恩斯坦函数的应用范围。
摘要：
我们建立了随机薛维方程 $ H^ω = ϕ(-\mathcal{L}) + V^ω $ 在平面无界嵌套分形上积分态密度（IDS）的Lifshitz奇异性。这里，$\mathcal{L}$ 是分形上的拉普拉斯算子，$ϕ$ 是一个单调运算函数，具有温和的正则性，而 $V^ω$ 是一个泊松随机势能，具有充分规则的轮廓。我们工作的主要创新之处在于证明了研究 $V^ω$ 可以有效地减少到对某种合金型势能的分析，其中站点不再是经典 $\mathbb{Z}^d$ 情况下的晶格点，而是分形复体。这一观察使我们能够应用一种新的方法，该方法在泊松随机场的设置中是新颖的，允许我们处理广泛类别的伯恩斯坦函数 $ϕ$。特别是，它涵盖了 $ϕ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$，$\vartheta \in (0,d_w)$，$m>0$，对应于相对论模型的情况，此前已知方法在分形上无法处理。
英文摘要：
We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators \[ H^ω = ϕ(-\mathcal{L}) + V^ω \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $ϕ$ is an operator monotone function with mild regularity, and $V^ω$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^ω$ can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical $\mathbb{Z}^d$ case, but fractal complexes. This observation enables us to apply an approach, new in the setting of Poissonian random fields, which allows us to treat a broad class of Bernstein functions $ϕ$. In particular, it covers the case $ϕ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.
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