Jacobi matrices with absolutely continuous spectrum

• Published in: Proceedings of the American Mathematical Society Vol. 127; no. 3; pp. 791 - 800
• Main Authors: Janas, Jan, Naboko, Serguei
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-03-1999
• Subjects: Continuous spectra; Mathematics; Matrices; Jacobi matrix; absolutely continuous spectrum; asymptotics behaviour
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-99-04586-4

Let J be a Jacobi matrix defined in l^2 as Re W, where W is a unilateral weighted shift with nonzero weights \lambda_k such that \lim_k \lambda_k = 1. Define the seqences: \varepsilon_k:= \frac{\lambda_{k-1}}{\lambda_k} -1, \delta_k:= \frac{\lambda_k -1}{\lambda_k}, \eta_k:= 2 \delta_k + \varepsilon_k. If \varepsilon_k = O(k^{-\alpha}) , \eta_k = O(k^{-\gamma}), \frac{2}{3}< \alpha \leq \gamma, \alpha + \gamma > 3/2 and \gamma > 3/4, then J has an absolutely continuous spectrum covering (-2,2). Moreover, the asymptotics of the solution Ju = \lambda u, \lambda \in \mathbb{R} is also given.