Skip to main content
SpringerLink
Log in

Inverse Resonance Problem for Jacobi Operators on a Half-Lattice

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite Jacobi matrices and theory of polynomials. We determine forbidden domains for resonances and maximal possible multiplicities of real and complex resonances.

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.
Fig. 2.
Fig. 3.

Similar content being viewed by others

References

  1. A. Aptekarev and E. Nikishin, “The Scattering Problem for a Discrete Sturm–Liouville Operator”, Sb. Math., 49:2 (1984), 325–355.

    Article  MATH  Google Scholar 

  2. T. Aktosun, V. Papanicolaou, and A. Rivero, “Darboux Transformation for the Discrete Schrodinger Equation”, Electronic Journal of Differential Equations, 112 (2019), 1–34.

    MathSciNet  MATH  Google Scholar 

  3. A. Badanin and E. Korotyaev, “Resonances of 4-th Order Differential Operators”, Asymp. Anal., 111 (2019), 137–177.

    MATH  Google Scholar 

  4. M. Bledsoe, “Stability of the inverse Resonance Problem for Jacobi Operators”, Integral Equations Operator Theory, 74:4 (2012), 481–496.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. de Boor and G. H. Golub, “The Numerically Stable Reconstruction of a Jacobi Matrix from Spectral Data”, Linear Algebra Appl., 21:3, (1978), 245–260.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Brown, S. Naboko, and R. Weirakd, “The Inverse Resonance Problem for Jacobi Operators”, Bull. London Math. Soc., 37 (2005), 727–37.

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Brown, I. Knowles, and R. Weikard, “On the Inverse Resonance Problem”, J. London Math. Soc., 68:2 (2003), 383–401.

    Article  MathSciNet  MATH  Google Scholar 

  8. K. M. Case, “On Discrete Inverse Scattering Problems, II”, J. Math. Phys., 14 (1973), 916–920.

    Article  ADS  MathSciNet  Google Scholar 

  9. K. M. Case, “The Discrete Inverse Scattering Problem in One Dimension”, J. Math. Phys., 15 (1974), 143–146.

    Article  ADS  MathSciNet  Google Scholar 

  10. K. M. Case and S. C. Chiu, “The Discrete Version of the Marchenko Equations in the Inverse Scattering Problem”, J. Math. Phys., 14 (1973), 1643–1647.

    Article  ADS  MathSciNet  Google Scholar 

  11. K. M. Case and M. Kac, “A Discrete Version of the Inverse Scattering Problem”, J. Math. Phys., 14 (1973), 594–603.

    Article  ADS  MathSciNet  Google Scholar 

  12. T. Christiansen, “Resonances for Steplike Potentials: Forward and Inverse Results”, Trans. Amer. Math. Soc., 358:5 (1973), 2071–2089.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Damanik and B. Simon, “Jost Functions and Jost Solutions for Jacobi Matrices I. A Necessary and Sufficient Condition for Szegö Asymptotics”, Invent. Math., 165:1 (2006), 1–50.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. D. Damanik and B. Simon, “Jost Functions and Jost Solutions for Jacobi Matrices, II”, Decay and analyticity. Int. Math. Res. Not. Art., 2006 (2006), 19396.

    MathSciNet  MATH  Google Scholar 

  15. N. Firsova, “Resonances of the Perturbed Hill Operator with Exponentially Decreasing Extrinsic Potential”, Mathematical Notes, 36:5 (1984), 854–861.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Froese, “Asymptotic Distribution of Resonances in One Dimension”, J. Diff. Eq., 137:2 (1997), 251–272.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. R. Froese and I. Herbst, “Resonances in the One Dimensional Stark Effect in the Limit of Small Field, SchrÖdinger Operators”, Spectral Analysis and Number Theory, (2021), 133–167.

    MATH  Google Scholar 

  18. G. S. Guseinov, “The Inverse Problem of Scattering Theory for a Second-Order Difference Equation on the Whole Axis”, Soviet Math. Dokl., 17 (1976), 1684–1688.

    MATH  Google Scholar 

  19. G. S. Guseinov, “Scattering Problem for the Infinite Jacobi Matrix”, Izv. Akad. Nauk Arm. SSR, Mat., 12 (1977), 365–379.

    MathSciNet  Google Scholar 

  20. F. Gesztesy and B. Simon, “M-Functions and Inverse Spectral Analysis For Finite and Semi-Infinite Jacobi Matrices”, J. Anal. Math., 73 (1997), 267–297.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Hitrik, “Bounds on Scattering Poles in One Dimension”, Comm. Math. Phys., 208:2 (1999), 381–411.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. H. Hochstadt, “On the Construction of a Jacobi Matrix From Spectral Data”, Linear Algebra Appl., 8 (1974), 435–446.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2012.

    Book  Google Scholar 

  24. A. Iantchenko and E. Korotyaev, “Schrodinger Operator on the Zigzag Half-Nanotube in Magnetic Field”, Math. Model. Nat. Phenom., 5:4 (2010), 175–197.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Iantchenko and E. Korotyaev, “Periodic Jacobi Operator with Finitely Supported Perturbation on the Half-Lattice”, Inverse Problems, 27:11, 115003, 26 pp. (2011).

    MathSciNet  MATH  Google Scholar 

  26. A. Iantchenko and E. Korotyaev, “Resonances for Periodic Jacobi Operators with Finitely Supported Perturbations”, J. Math. Anal. Appl., 388:2 (2012), 1239–1253.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Iantchenko and E. Korotyaev, “Periodic Jacobi Operator with Finitely Supported Perturbations: The Inverse Resonance Problem”, J. Diff. Equations, 252:3 (2012), 2823–2844.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. H. Isozaki and E. Korotyaev, “Inverse Problems, Trace Formulae for Discrete Schrödinger Operators”, Annales Henri Poincare, 13:4 (2012), 751–788.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. R. Killip and B. Simon, “Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory”, Annals of Mathematics. Second Series, 158:1 (2003), 253–321.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Korotyaev, “Inverse Resonance Scattering on the Half Line”, Asymptot. Anal., 37:3-4 (2004), 215–226.

    MathSciNet  MATH  Google Scholar 

  31. E. Korotyaev, “Stability for Inverse Resonance Problem”, Int. Math. Res. Not., 73 (2004), 3927–3936.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. Korotyaev, “Inverse Resonance Scattering on the Real Line”, Inverse Problems, 21:1 (2005), 325–341.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. E. Korotyaev, “Inverse Resonance Scattering for Jacobi Operators”, Russ. J. Math. Phys., 18 (2011), 427–439.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Korotyaev, “Resonance Theory for Perturbed Hill Operator”, Asymp. Anal., 74:3-4 (2011), 199–227.

    MathSciNet  MATH  Google Scholar 

  35. E. Korotyaev, “Estimates of 1D Resonances in Terms of Potentials”, Journal d’Analyse Math., 130 (2016), 151–166.

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Korotyaev, “Resonances for 1D Stark Operators”, Journal of Spectral Theory, 7:3 (2017), 699–732.

    Article  MathSciNet  MATH  Google Scholar 

  37. E. Korotyaev, “Resonances of Third Order Differential Operators”, Journal of Math. Analysis and Appl., 478:1 (2019), 82–107.

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Korotyaev, “Discretization of Inverse Scattering on a Half Line, in press”, Math. Nachrichten, (2023).

    Google Scholar 

  39. E. Korotyaev Eigenvalues of Schrödinger Operators on Finite and Infinite Intervals, Math. Nachrichten, 294:11 (2021), 2188–2199.

    Article  MathSciNet  Google Scholar 

  40. E. Korotyaev and D. Mokeev, “Inverse Resonance Scattering for Dirac Operators on the Half-Line”, Analysis and Mathematical Physics, 11:1 Paper No. 32, 26 pp (2021).

    MathSciNet  MATH  Google Scholar 

  41. E. Korotyaev and D. Mokeev, “Inverse Resonance Scattering for Massless Dirac Operators on the Real Line”, Asympt. Anal., 132 (2023), 83–130.

    MathSciNet  MATH  Google Scholar 

  42. E. Korotyaev and K. Schmidt, “On the Resonances and Eigenvalues for a 1D Half-Crystal with Localized Impurity”, J. Reine Angew. Math., 670 (2012), 217–248.

    MathSciNet  MATH  Google Scholar 

  43. R. Kozhan, “Finite Range Perturbations of Finite Gap Jacobi and CMV Operators”, Advances in Mathematics, 301 (2016), 204–226.

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Marletta, S. Naboko, R. Shterenberg, and R. Weikard, “On the Inverse Resonance Problem for Jacobi Operators – Uniqueness and Stability”, J. Anal. Math., 117 (2012), 221–248.

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Marletta, R. Shterenberg, and R. Weikard, “On the Inverse Resonance Problem for Schrödinger Operators”, Commun. Math. Phys.,, 295 (2010), 465–484.

    Article  ADS  MATH  Google Scholar 

  46. M. Marletta and R. Weikard, “Stability for the Inverse Resonance Problem for a Jacobi Operator with Complex Potential”, Inverse Problems, 23 (2007), 1677–1688.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. S. Novikov, S. Manakov, L. Pitaevski, and V. Zakharov, Theory of Solitons. The Inverse Scattering Method, Consultants Bureau [Plenum], New York, 1984.

    Google Scholar 

  48. B. Simon, “Resonances in One Dimension and Fredholm Determinants”, J. Funct. Anal., 178:2 (2000), 396–420.

    Article  MathSciNet  MATH  Google Scholar 

  49. B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society Colloquium Publications, vol. 54, Part 1, American Mathematical Society, Providence, RI, 2005.

    Google Scholar 

  50. B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. American Mathematical Society Colloquium Publications,, vol. 54, Part 2, American Mathematical Society, Providence, RI, 2005.

    Google Scholar 

  51. G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72, AMS, Providence, RI, 2000.

    MATH  Google Scholar 

  52. M. Toda, Theory of Nonlinear Lattices, 2nd ed. Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  53. P. van Moerbeke, “The Spectrum of Jacobi Matrices”, Invent. Math., 37:1 (1976), 45-81.

    ADS  MathSciNet  MATH  Google Scholar 

  54. R. Weikard and M. Zinchenko, “The Inverse Resonance Problem for CMV Operators”, Inverse Problems, 26:5 055012 (2010).

    MathSciNet  MATH  Google Scholar 

  55. M. Zworski, “A Remark on Isopolar Potentials”, SIAM J. Math. Anal., 32 (2001), 1324–1326.

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Zworski, “Distribution of Poles for Scattering on the Real Line”, J. Funct. Anal., 73 (1987), 277–296.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation, grant no 23-21-00023.

Author information

Authors and Affiliations

  1. Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia

    E. Korotyaev & E. Leonova

Authors
  1. E. Korotyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Leonova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Korotyaev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Korotyaev, E., Leonova, E. Inverse Resonance Problem for Jacobi Operators on a Half-Lattice. Russ. J. Math. Phys. 30, 320–344 (2023). https://doi.org/10.1134/S1061920823030056

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920823030056

Search

Navigation