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  • J. Janas (a1) and S. Naboko (a2)


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.



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[1]Barbaroux, J. M., Combes, J.-M. and Hislop, P. D., Localization near band edges for random Schrödinger operators. Helv. Phys. Acta 70 (1997), 1643.
[2]Berezanskii, Yu. M., Expansions in Eigenfunctions of Selfadjoint Operators (Translations of Mathematical Monographs 17), American Mathematical Society (Providence, RI, 1968).
[3]Boutet de Monvel, A., Janas, J. and Naboko, S., Unbounded Jacobi matrices with a few gaps in the essential spectrum. Constructive examples. Integral Equations Operator Theory 69(2) (2011), 151170.
[4]Demko, S., Inverses of band matrices and local convergence of spline projections. Siam J. Numer. Anal. 14 (1977), 616619.
[5]Dombrowski, J., Eigenvalues and spectral gaps related to periodic perturbation of Jacobi matrices. Oper. Theory Adv. Appl. 154 (2004), 91100.
[6]Elaydi, S. N., An Introduction to Difference Equations, Springer (New York, 1999).
[7]Germinet, F. and Klein, A., Operator kernel estimates for functions of generalized Schrödinger operators. Proc. Amer. Math. Soc. 131(2) (2002), 911920.
[8]Jaffard, S., Proprietes des matrices “bien localisees” pres de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(5) (1990), 461476.
[9]Janas, J. and Moszyński, M., Spectral properties of Jacobi matrices by asymptotic analysis. J. Approx. Theory 120(2) (2003), 309336.
[10]Janas, J. and Naboko, S., Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes. Oper. Theory Adv. Appl. 127 (2001), 387397.
[11]Janas, J., Naboko, S. and Stolz, G., Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods. J. Comput. Appl. Math. 171(1–2) (2004), 265276.
[12]Janas, J., Naboko, S. and Stolz, G., Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not. 4 (2009), 736764.
[13]Kato, T., Perturbation Theory for Linear Operators (Classics in Mathematics), Springer (Berlin, 1995), reprint of the 1980 edition.
[14]Naboko, S. and Simonov, S., Spectral analysis of a class of Hermitian Jacobi matrices in a critical (double root) hyperbolic case. Proc. Edinb. Math. Soc. (2) 53(1) (2010), 239254.
[15]Shubin, M. A., Pseudodifference operators and their Green functions. Sibirsk. Mat. Zh. 49 (1985), 652671 (in Russian).
[16]Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Lattices (Mathematical Surveys and Monographs 72), American Mathematical Society (Providence, RI, 2000).
[17]Yafaev, D., Exponential decay of eigenfunctions of first order systems. Preprint, 2007, arXiv:math/0701303v1.
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