Skip to main content
SpringerLink
Log in

On the Instability of the Essential Spectrum for Block Jacobi Matrices

  • Published:
Constructive Approximation Aims and scope

Abstract

We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing Co., New York (1965)

    MATH  Google Scholar 

  2. Benzaid, Z., Lutz, D.A.: Asymptotic representation of solutions of perturbed systems of linear difference equations. Stud. Appl. Math. 77(3), 195–221 (1987)

    Article  MathSciNet  Google Scholar 

  3. Berezanskii, Yu. M.: Expansions in eigenfunctions of selfadjoint operators. Translations AMS, vol. 17. AMS, Providence, RI (1968)

  4. Birman, M., Solomjak, M.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series). D. Reidel Publishing, Dordrecht (1987)

    Google Scholar 

  5. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company Inc, New York (1955)

    MATH  Google Scholar 

  6. Damanik, D., Naboko, S.: Unbounded Jacobi matrices at critical coupling. J. Approx. Theory 145(2), 221–236 (2007)

    Article  MathSciNet  Google Scholar 

  7. Damanik, D., Pushnitskii, A., Simon, B.: The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory 4, 1–85 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2005)

    Google Scholar 

  9. Gilbert, D.J., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128(1), 30–56 (1987)

    Article  MathSciNet  Google Scholar 

  10. Janas, J., Moszyński, M.: Spectral properties of Jacobi matrices by asymptotic analysis. J. Approx. Theory 120(2), 309–336 (2003)

    Article  MathSciNet  Google Scholar 

  11. Janas, J., Moszyński, M.: Spectral analysis of unbounded Jacobi operators with oscillating entries. Stud. Math. 209(2), 107–133 (2012)

    Article  MathSciNet  Google Scholar 

  12. Janas, J., Naboko, S.: Multithreshold spectral phase transitions for a class of Jacobi matrices. In: Recent Advances in Operator Theory (Gröningen, 1998), pp. 267–285, Oper. Theory Adv. Appl., 124, Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  13. Janas, J., Naboko, S.: Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes. In: Recent Advances in Operator Theory and Related Topics (Szeged, 1999), pp. 387–397, Oper. Theory Adv. Appl., 127, Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  14. Janas, J., Naboko, S.: Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries. J. Funct. Anal. 191(2), 318–342 (2002)

    Article  MathSciNet  Google Scholar 

  15. Janas, J., Naboko, S.: On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach. J. Differ. Equ. Appl. 21(11), 1103–1118 (2015)

    Article  MathSciNet  Google Scholar 

  16. Janas, J., Naboko, S., Sheronova, E.: Asymptotic behavior of generalized eigenvectors of Jacobi matrices in the critical (double root) case. Z. Anal. Anwend. 28(4), 411–430 (2009)

    Article  MathSciNet  Google Scholar 

  17. Janas, J., Naboko, S., Stolz, G.: Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods. J. Comput. Appl. Math. 171(1–2), 265–276 (2004)

    Article  MathSciNet  Google Scholar 

  18. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). Reprint of the 1980 edition

    Book  Google Scholar 

  19. Khan, S., Pearson, D.B.: Subordinacy and spectral theory for infinite matrices. Helv. Phys. Acta 65(4), 505–527 (1992)

    MathSciNet  Google Scholar 

  20. Naboko, S., Pchelintseva, I., Silva, L.: Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis. J. Approx. Theory 161(1), 314–336 (2009)

    Article  MathSciNet  Google Scholar 

  21. Naboko, S., 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), 239–254 (2010)

    Article  MathSciNet  Google Scholar 

  22. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. Academic Press Inc, New York (1980)

    MATH  Google Scholar 

  23. Teschl, G.: Jacobi operators and completely integrable nonlinear lattices. AMS Surveys and Monographs, vol. 72. AMS, Providence, RI (2000)

  24. Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)

    Book  Google Scholar 

  25. Weidmann, J.: Linear Operators in Hilbert Spaces Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)

    Book  Google Scholar 

  26. Weidmann, J. Lineare Operatoren in Hilberträumen. Teil 1. (German) [Linear operators in Hilbert spaces. Part 1] Grundlagen. [Foundations] Mathematische Leitfäden. [Mathematical Textbooks] Teubner, Stuttgart (2000)

    Google Scholar 

Download references

Acknowledgements

Both authors are close friends of Leonid Golinskii, and SK is Leonid’s longtime collaborator. It is our great pleasure to dedicate this article to Leonid’s 65-th anniversary. We wish Leonid many more years of blossoming scientific activity, as well as many nice mathematical results in the distinctive “LG soft power” style. We would like to thank Prof. C. Tretter for helpful remarks on the subject of the paper. SN is partially supported by Grants RSF 15-11-30007, RFBR 16-11-00443a, NCN 2013/BST/04319, K. and A. Wallenberg grant of Wallenberg Association and Royal Swedish Academy of Sciences, and by the project SPbGU N\(^{{\underline{\circ }}}\)11.42.1071.2016. The research presented in the paper was done during his stay at University of Bordeaux in the framework of the “IdEx International Scholars” program. SN would like to acknowledge the financial support of the program, as well as the hospitality of the Institute of Mathematics of Bordeaux where he was hosted.

Author information

Authors and Affiliations

  1. IMB, CNRS, Université de Bordeaux, 351 ave. de la Libération, 33405, Talence Cedex, France

    S. Kupin

  2. Physics Institute, St. Petersburg State University, Ulyanovskaya str. 1, St. Peterhof, St. Petersburg, Russia, 198-904

    S. Naboko

Authors
  1. S. Kupin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Naboko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Kupin.

Additional information

Communicated by Sergey Denisov.

To Leonid Golinskii on the occasion of his 65-th anniversary.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kupin, S., Naboko, S. On the Instability of the Essential Spectrum for Block Jacobi Matrices. Constr Approx 48, 473–500 (2018). https://doi.org/10.1007/s00365-018-9436-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-018-9436-4

Keywords

Mathematics Subject Classification

Search

Navigation