Skip to main content
SpringerLink
Log in

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates the spectral density of a matrix to the asymptotics of orthogonal polynomials associated with it.

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

Fig. 1.

Similar content being viewed by others

Notes

  1. The branch of the square root should be chosen so that its values are positive for positive \(\lambda\).

References

  1. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, Edinburgh, 1965.

    MATH  Google Scholar 

  2. N. I. Akhiezer and I. M. Glazman, Theory of Operators in Hilbert Space, Frederick Ungar, New York, 1963.

    MATH  Google Scholar 

  3. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  4. E. C. Titchmarsh, Eigenfunction Expansions, vol. Part 1,, Clarendon Press, Oxford, 1962.

    MATH  Google Scholar 

  5. W. Van Assche and J. S. Geronimo, “Asymptotics for orthogonal polynomials on and off the essential spectrum”, J. Approx. Theory, 55:2 (1988), 220–231.

    Article  MathSciNet  Google Scholar 

  6. A. I. Aptekarev and J. S. Geronimo, “Measures for orthogonal polynomials with unbounded recurrence coefficients”, J. Approx. Theory, 207 (2016), 339–347.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. S. Bodine and D. A. Lutz, Asymptotic Integration of Differential and Difference Equations, Springer, Cham, 2015.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. W. A. Harris, Jr. and D. A. Lutz, “Asymptotic integration of adiabatic oscillators”, J. Math. Anal. Appl., 51 (1975), 76–93.

    Article  MathSciNet  Google Scholar 

  12. E. Ianovich, “On one condition of absolutely continuous spectrum for self-adjoint operators and its applications”, Opuscula Math., 38:5 (2018), 699–718.

    Article  MathSciNet  Google Scholar 

  13. J. Janas, “The asymptotic analysis of generalized eigenvectors of some Jacobi operators. Jordan box case”, J. Difference Equ. Appl., 12:6 (2006), 597–618.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. J. Janas and S. Naboko, “Multithreshold spectral phase transitions for a class of Jacobi matrices”, Recent Advances in Operator Theory (Groningen, 1998), Oper. Theory Adv. Appl., 124 Birkäuser, Basel, 2001, 267–285.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. J. Janas and S. Simonov, “Weyl–Titchmarsh type formula for discrete Schrödinger operator with Wigner–von Neumann potential”, Studia Math., 201:2 (2010), 167–189.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  20. R.-J. Kooman, “An asymptotic formula for solutions of linear second-order difference equations with regularly behaving coefficients”, J. Difference Equ. Appl., 13:11 (2007), 1037–1049.

    Article  MathSciNet  Google Scholar 

  21. P. Kurasov and S. Simonov, “Weyl–Titchmarsh-type formula for periodic Schrödinger operator with Wigner–von Neumann potential”, Proc. Roy. Soc. Edinburgh, Sect. A, 143 (2013), 401–425.

    Article  MathSciNet  Google Scholar 

  22. A. Máté, P. Nevai, and V. Totik, “Asymptotics for orthogonal polynomials defined by a recurrence relation”, Constr. Approx., 1:3 (1985), 231–248.

    Article  MathSciNet  Google Scholar 

  23. S. Naboko and S. Simonov, “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.

    Article  MathSciNet  Google Scholar 

  24. S. Naboko and S. Simonov, “Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential”, Math. Proc. Cambridge Philos. Soc., 153:1 (2012), 33–58.

    Article  MathSciNet  Google Scholar 

  25. S. Naboko and M. Solomyak, “On the absolutely continuous spectrum in a model of an irreversible quantum graph”, Proc. London Math. Soc. (3), 92:1 (2006), 251–272.

    Article  MathSciNet  Google Scholar 

  26. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.

    MATH  Google Scholar 

  27. L. O. Silva, “Uniform and smooth Benzaid–Lutz type theorems and applications to Jacobi matrices”, Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl., 174 Birkhäuser, Basel, 2007, 173–186.

    Article  MathSciNet  Google Scholar 

  28. S. Simonov, “An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights”, Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl., 174 Birkhäuser, Basel, 2007, 187–203.

    Article  MathSciNet  Google Scholar 

  29. S. Simonov, “Weyl–Titchmarsh type formula for Hermite operator with small perturbation”, Opuscula Math., 29:2 (2009), 187–207.

    Article  MathSciNet  Google Scholar 

  30. S. Simonov, “Zeroes of the spectral density of discrete Schrödinger operator with Wigner–von Neumann potential”, Integral Equations Operator Theory, 73:3 (2012), 351–364.

    Article  MathSciNet  Google Scholar 

  31. S. Simonov, “Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential”, Math. Z., 284:1–2 (2016), 335–411.

    Article  MathSciNet  Google Scholar 

  32. G. Świderski, “Spectral properties of unbounded Jacobi matrices with almost monotonic weights”, Constr. Approx., 44:1 (2016), 141–157.

    Article  MathSciNet  Google Scholar 

  33. G. Świderski and B. Trojan, “Periodic perturbations of unbounded Jacobi matrices”, J. Approx. Theory, 216 (2017), 38–66.

    Article  MathSciNet  Google Scholar 

  34. G. Świderski, “Periodic perturbations of unbounded Jacobi matrices”, J. Approx. Theory, 216 (2017), 67–85.

    Article  MathSciNet  Google Scholar 

  35. G. Świderski, “Periodic perturbations of unbounded Jacobi matrices”, J. Approx. Theory, 233 (2018), 1–36.

    Article  MathSciNet  Google Scholar 

  36. R. Szwarc, “Absolute continuity of spectral measure for certain unbounded Jacobi matrices”, Advanced Problems in Constructive Approximation, Internat. Ser. Numer. Math., 142 Birkhäuser, Basel, 2002, 255–262.

    Article  Google Scholar 

Download references

Funding

The first author was supported by the RFBR grant 19-01-00657A, by the Knut and Alice Wallenberg Foundation (Sections 1–3), and by the RScF grant 20-11-20032 (Section 4). He appreciates the hospitality of the Mittag-Leffler Institute, where a part of this work was done. The second author was supported by the RFBR grant 19-01-00565A (Sections 1–3) and by the RScF grant 20-11-20032 (Section 4).

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    S. N. Naboko & S. A. Simonov

  2. St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

    S. A. Simonov

Authors
  1. S. N. Naboko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Simonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. N. Naboko or S. A. Simonov.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 21–43 https://doi.org/10.4213/faa3857.

To the memory of M. Z. Solomyak

Translated by S. A. Simonov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naboko, S.N., Simonov, S.A. Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case. Funct Anal Its Appl 55, 94–112 (2021). https://doi.org/10.1134/S0016266321020027

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266321020027

Keywords

Search

Navigation