Abstract
We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)
Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer. Hafner Publishing Co., New York (1965)
Beckermann, B.: Complex Jacobi matrices. J. Comput. Appl. Math. 127(1–2), 17–65 (2001)
Berezanskiĭ, J. M.: Expansions in Eigenfunctions of Selfadjoint Operators. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, vol. 17. American Mathematical Society, Providence (1968)
Conway, J.B.: Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11, 2nd edn. Springer, New York (1978)
Davies, E.B.: Linear Operators and Their Spectra. Cambridge Studies in Advanced Mathematics, vol. 106. Cambridge University Press, Cambridge (2007)
Gasper, G., Rahman, M.: Basic Hypergeometric Series Encyclopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990). With a foreword by Richard Askey
Helffer, B.: Spectral Theory and Its Applications. Cambridge Studies in Advanced Mathematics, vol. 139. Cambridge University Press, Cambridge (2013)
Kato, T.: Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer, New York (1966)
Masson, D.R., Repka, J.: Spectral theory of Jacobi matrices in \(l^2({ Z})\) and the \({\rm su}(1,1)\) Lie algebra. SIAM J. Math. Anal. 22(4), 1131–1146 (1991)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1978)
Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137(1), 82–203 (1998)
Simon, B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120, 2nd edn. American Mathematical Society, Providence (2005)
Štampach, F., Šťovíček, P.: On the eigenvalue problem for a particular class of finite Jacobi matrices. Linear Algebra Appl. 434(5), 1336–1353 (2011)
Štampach, F., Šťovíček, P.: The characteristic function for Jacobi matrices with applications. Linear Algebra Appl. 438(11), 4130–4155 (2013)
Štampach, F., Šťovíček, P.: Special functions and spectrum of Jacobi matrices. Linear Algebra Appl. 464, 38–61 (2015)
Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72. American Mathematical Society, Providence (2000)
Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)
Wall, H.S.: Analytic Theory of Continued Fractions. D. Van Nostrand Company Inc, New York (1948)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Štampach, F. The Characteristic Function for Complex Doubly Infinite Jacobi Matrices. Integr. Equ. Oper. Theory 88, 501–534 (2017). https://doi.org/10.1007/s00020-017-2357-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-017-2357-y