We investigate the behavior near zero of the integrated density of states for random Schrödinger operators Φ(−Δ) + Vω in L2(ℝd), d ≥ 1, where Φ is a complete Bernstein function such that for some α ∈ (0, 2], one has Φ(λ) ≍ λα/2, λ ↘ 0, and Vω(x) =∑ i∈ℤdqi(ω)W(x −i) is a random nonnegative alloy-type potential with compactly supported single site potential W. We prove that there are constants C,C̃,D,D̃ > 0 such that −C ≤liminfλ↘0 λd/α |log Fq(Dλ)|log N(λ)andlimsupλ↘0 λd/α |log Fq(D̃λ)|log N(λ) ≤−C̃, where Fq is the common cumulative distribution function of the lattice random variables qi. For typical examples of Fq, the constants D and D̃ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others.

中文翻译：

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Lévy 算子驱动的连续 Anderson 模型的 Lifshitz 尾

我们研究了随机薛定谔算子的状态积分密度接近零的行为Φ(-Δ) + 五ω在大号2(ℝd),d ≥ 1， 在哪里Φ是一个完整的 Bernstein 函数，使得对于某些α ∈ (0, 2], 一个有Φ(λ) ≍ λα/2,λ ↘ 0， 和五ω(X) =∑ 一世∈ℤdq一世(ω)W(X -一世)是具有紧支撑单点势的随机非负合金型势W. 我们证明存在常数C,C̃,D,D̃ > 0这样 -C ≤限制λ↘0 λd/α |日志 Fq(Dλ)|日志 ñ(λ)和利姆苏普λ↘0 λd/α |日志 Fq(D̃λ)|日志 ñ(λ) ≤-C̃, 在哪里Fq是格随机变量的常见累积分布函数q一世. 对于典型的例子Fq,常数D和D̃可以从上面的陈述中消除。我们结合概率和分析方法，允许以统一的方式处理拉普拉斯算子的大类算子单调函数。此类包括局部和非局部动力学项，例如拉普拉斯算子、其分数幂、准相对论哈密顿量和许多其他项。