关于Prüfer角的uniform distribution及其对随机稀疏扰动Jacobi矩阵谱转移的意义
On the uniform distribution of the Prüfer angles and its implication to a sharp spectral transition of Jacobi matrices with randomly sparse perturbations
摘要：
在目前的工作中，我们考虑非对角雅可比矩阵的稀疏扰动的位置不确定性。证明了（定理3.2）对于除有理数集以外的所有实数\in [0，π]，以及对于几乎所有ω，关于{-j，.的一致测度的乘积ν= \prod_{j\geq 1} ν_{j}的Prüfer角序列（θ_{k}^ω）_{kPDMATHABMM 1}是u.d mod π，j}。与纯点谱的改进准则（引理4.1）一起，这为Zlatos（J. Funct. Anal. \textbf{207}，216-252（2004））：纯点（p.p）谱和奇异连续（s.c.）分别在集合A_{sc}和A_{pp}之外的关于本质谱[-2，2]彼此互补的集合上的谱，两者都是零勒贝格测度（定理2.4）。我们的方法允许一个明确的表征A_{pp}，这是被视为也是密集的p. p.型，从而证明了该光谱是唯一的纯点上的一个子集的本质谱。
英文摘要：
In the present work we consider off-diagonal Jacobi matrices with uncertainty in the position of sparse perturbations. We prove (Theorem 3.2) that the sequence of Prüfer angles (θ_{k}^ω)_{k\geq 1} is u.d mod πfor all ϕ\in [0,π] with exception of the set of rational numbers and for almost every ωwith respect to the product ν=\prod_{j\geq 1}ν_{j} of uniform measures on {-j,...,j}. Together with an improved criterion for pure point spectrum (Lemma 4.1), this provides a simple and natural alternative proof of a result of Zlatos (J. Funct. Anal. \textbf{207}, 216-252 (2004)): the existence of pure point (p.p) spectrum and singular continuous (s.c.) spectra on sets complementary to one another with respect to the essential spectrum [-2,2], outside sets A_{sc} and A_{pp}, respectively, both of zero Lebesgue measure (Theorem 2.4). Our method allows for an explicit characterization of A_{pp}, which is seen to be also of dense p.p. type, and thus the spectrum is proved to be exclusively pure point on one subset of the essential spectrum.
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