由加倍映射驱动的一维散度型雅可比算子
One-Dimensional Divergence-Type Jacobi Operators Driven by The Doubling Map
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中文总结 AI 辅助
本文研究由加倍映射生成系数的一维散度型雅可比算子，证明其几乎必然本质谱性质，得到积分态密度、Lyapunov指数的相关渐近与连续性结果，还将薛定谔算子的小耦合结果推广至该类算子并得到安德森局域化。
中文摘要 AI 辅助
我们研究定义在ℓ²(ℤ≥₀)上、由加倍映射生成系数的一维散度-梯度型雅可比算子，其系数为aₙ(x)=a(2ⁿx mod 1)。对于连续正采样函数，我们证明了几乎必然本质谱是包含谱底的区间。在假设a∈C¹(𝕋)的条件下，我们得到了积分态密度的平方根渐近行为，并给出了Lyapunov指数在E→0⁺时的小能量展开式。Lyapunov指数的主项为线性正项，当且仅当采样函数满足非退化条件时成立。在该条件下，我们对转移矩阵证明了大偏差估计，其显式速率依赖于E。作为推论，我们得到了谱底附近Lyapunov指数与积分态密度的局部Hölder连续性，以及安德森局域化。部分结果将Chulaevsky–Spencer和Bourgain–Schlag关于加倍映射生成的薛定谔算子的小耦合结果推广到了散度-梯度型算子。论证过程借鉴了上述工作的方法，但需要对小能量参数进行一致控制，并结合反映散度-梯度结构的精细相关估计。
英文摘要
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local Hölder continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schrödinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
