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ESTIMATES OF GENERALIZED EIGENVECTORS OF HERMITIAN JACOBI MATRICES WITH A GAP IN THE ESSENTIAL SPECTRUM

Part of: Special classes of linear operators

Published online by Cambridge University Press:  14 May 2012

J. Janas  and
S. Naboko
J. Janas
Affiliation:
Institute of Mathematics PAN, 31-027 Cracow, Poland (email: [email protected])
S. Naboko
Affiliation:
Department of Mathematical Physics, Institute of Physics, St. Petersburg, 198904, Russia (email: [email protected])
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Abstract

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.

MSC classification

Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 191 - 212
Copyright
Copyright © University College London 2012

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References

[1]Barbaroux, J. M., Combes, J.-M. and Hislop, P. D., Localization near band edges for random Schrödinger operators. Helv. Phys. Acta 70 (1997), 16–43.Google Scholar
[2]Berezanskii, Yu. M., Expansions in Eigenfunctions of Selfadjoint Operators (Translations of Mathematical Monographs 17), American Mathematical Society (Providence, RI, 1968).CrossRefGoogle Scholar
[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), 151–170.CrossRefGoogle Scholar
[4]Demko, S., Inverses of band matrices and local convergence of spline projections. Siam J. Numer. Anal. 14 (1977), 616–619.CrossRefGoogle Scholar
[5]Dombrowski, J., Eigenvalues and spectral gaps related to periodic perturbation of Jacobi matrices. Oper. Theory Adv. Appl. 154 (2004), 91–100.Google Scholar
[6]Elaydi, S. N., An Introduction to Difference Equations, Springer (New York, 1999).CrossRefGoogle Scholar
[7]Germinet, F. and Klein, A., Operator kernel estimates for functions of generalized Schrödinger operators. Proc. Amer. Math. Soc. 131(2) (2002), 911–920.CrossRefGoogle Scholar
[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), 461–476.CrossRefGoogle Scholar
[9]Janas, J. and Moszyński, M., Spectral properties of Jacobi matrices by asymptotic analysis. J. Approx. Theory 120(2) (2003), 309–336.CrossRefGoogle Scholar
[10]Janas, J. and Naboko, S., Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes. Oper. Theory Adv. Appl. 127 (2001), 387–397.Google Scholar
[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), 265–276.CrossRefGoogle Scholar
[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), 736–764.Google Scholar
[13]Kato, T., Perturbation Theory for Linear Operators (Classics in Mathematics), Springer (Berlin, 1995), reprint of the 1980 edition.CrossRefGoogle Scholar
[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), 239–254.CrossRefGoogle Scholar
[15]Shubin, M. A., Pseudodifference operators and their Green functions. Sibirsk. Mat. Zh. 49 (1985), 652–671 (in Russian).Google Scholar
[16]Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Lattices (Mathematical Surveys and Monographs 72), American Mathematical Society (Providence, RI, 2000).Google Scholar
[17]Yafaev, D., Exponential decay of eigenfunctions of first order systems. Preprint, 2007, arXiv:math/0701303v1.CrossRefGoogle Scholar