Skip to main content
SpringerLink
Log in

Green Matrix Estimates of Block Jacobi Matrices I: Unbounded Gap in the Essential Spectrum

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

This work deals with decay bounds for Green matrices and generalized eigenvectors of block Jacobi matrices when the real part of the spectral parameter lies in an infinite gap of the operator’s essential spectrum. We consider the cases of commutative and noncommutative matrix entries separately. An example of a block Jacobi operator with noncommutative entries and nonnegative essential spectrum is given to illustrate the results.

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., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Translated from the Russian and with a Preface by Merlynd Nestell, Reprint of the 1961 and 1963 Translations, Two Volumes Bound as One. Dover Publications Inc., New York (1993)

    MATH  Google Scholar 

  2. Barbaroux, J.M., Combes, J.M., Hislop, P.D.: Localization near band edges for random Schrödinger operators. Papers Honouring the 60th Birthday of Klaus Hepp and of Walter Hunziker, Part II (Zürich, 1995). Helv. Phys. Acta 70(1–2), 16–43 (1997)

    MathSciNet  Google Scholar 

  3. Berezans’kiĭ, 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, R.I. (1968)

    Google Scholar 

  4. Birman, M.S., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series). Translated from the 1980 Russian Original by S. Khrushchëv, V. Peller. D. Reidel Publishing Co., Dordrecht (1987)

    Google Scholar 

  5. Combes, J.M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, 251–270 (1973)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators. J. Eur. Math. Soc. (JEMS) 16(5), 909–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230(1), 184–221 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gebert, M., Müller, P.: Localization for random block operators. In: Demuth, M., Kirsch, W. (eds.) Mathematical Physics, Spectral Theory and Stochastic Analysis. Operator Theory: Advances and Applications, vol. 232, pp. 229–246. Birkhäuser, Basel (2013)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Janas, J., Naboko, S.: Estimates of generalized eigenvectors of Hermitian Jacobi matrices with a gap in the essential spectrum. Mathematika 59(1), 191–212 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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  MATH  Google Scholar 

  13. Janas, J., Naboko, S., Stolz, G.: Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not. IMRN 4, 736–764 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Kato, T.: Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften, Band 132, 2nd edn. Springer, Berlin (1976)

    Google Scholar 

  15. Kirsch, W.: An invitation to random Schrödinger operators. In: Disertori, M., et al. (eds.) Random Schrödinger Operators. Panorama et Synthèses. With an Appendix by Frédéric Klopp, vol. 25, pp. 1–119. Société Mathématique de France, Paris (2008).

  16. Kotani, S., Simon, B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Commun. Math. Phys. 119(3), 403–429 (1988)

    Article  MATH  Google Scholar 

  17. 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  MATH  Google Scholar 

  18. Sadel, C.: Spectral theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models. J. Funct. Anal. 274(8), 2205–2244 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sadel, C., Schulz-Baldes, H.: Spectral averaging techniques for Jacobi matrices with matrix entries. J. Phys. A 42(18), 185204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Świderski, G.: Spectral properties of block Jacobi matrices. Preprint, arXiv:1705.06138 (2017)

Download references

Acknowledgements

JJ and SN have been supported by the National Science Centre—Poland, Grant No. 2013/09/B/ST1/04319. SN was also supported by Grant RFBR 16-01-00443-a. LOS has been supported by UNAM-DGAPA-PAPIIT IN110818 and SEP-CONACYT CB-2015 254062. The authors thank the anonymous referee whose pertinent comments led to an improved presentation of this work.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences (PAN), Ul. Sw. Tomasza 30, 31-027, Kraków, Poland

    Jan Janas

  2. Department of Mathematical Physics Institute of Physics, St. Petersburg State University, Ulyanovskaya 1, St., Petersburg, Russia, 198904

    Sergey Naboko

  3. Departamento de Física Matemática Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Luis O. Silva

Authors
  1. Jan Janas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey Naboko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luis O. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis O. Silva.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Janas, J., Naboko, S. & Silva, L.O. Green Matrix Estimates of Block Jacobi Matrices I: Unbounded Gap in the Essential Spectrum. Integr. Equ. Oper. Theory 90, 49 (2018). https://doi.org/10.1007/s00020-018-2476-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-018-2476-0

Mathematics Subject Classification

Keywords

Search

Navigation