On the uniform distribution of the Pr\'{u}fer angles and its implication to a sharp spectral transition of Jacobi matrices with randomly sparse perturbations
| Autor: | Carvalho, S. L., Marchetti, D. H. U., Wreszinski, W. F. |
|---|---|
| Rok vydání: | 2010 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 Pr\"ufer angles (\theta_{k}^{\omega})_{k\geq 1} is u.d mod \pi for all \phi \in [0,\pi] with exception of the set of rational numbers and for almost every \omega with respect to the product \nu =\prod_{j\geq 1}\nu_{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. Comment: Submitted to Journal Functional Analysis on August 7, 2009; Submitted to Transactions of the American Mathematical Society on September 10, 2010; Submitted to Journal of Spectral Theory on March 7, 2011; 21 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|