Estimates of generalized eigenvectors of hermitian jacobi matrices with a gap in the essential spectrum

摘要

In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S.Naboko and G.Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int.Math.Res.Not. 4 (2009), 736-764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.