On the uniform distribution of the Prüfer angles and its implication to a sharp spectral transition of Jacobi matrices with randomly sparse perturbations

Abstract

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.