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Inverse resonance scattering for Jacobi operators

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Abstract

The Jacobi operator (Jf) n = a n−1 f n−1 +a n f n+1 + b n f n on ℤ with real finitely supported sequences (a n − 1) n∈ℤ and (b n ) n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a n , b n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a n , b n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. 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), 1–50 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. D. Damanik and B. Simon, “Jost Functions and Jost Solutions for Jacobi Matrices, II. Decay and Analyticity,” Int. Math. Res. Not. Art. ID 19396, 32 (2006).

  8. P. Deift and E. Trubowitz, “Inverse Scattering on the Line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  9. L. D. Faddeev, “Properties of the S-Matrix of the One-Dimensional Schrödinger Equation,” Tr. Mat. Inst. Steklova 73, 314–336 (1964) [Amer. Math. Soc. Transl. (2) 65, 139–166 (1967)].

    MATH  MathSciNet  Google Scholar 

  10. L. D. Faddeev and V. E. Zakharov “Korteweg-de Vries Equation is a Completely Integrable Hamiltonian System,” Funktsional. Anal. Prilozhen. 5(4), 18–27 (1971).

    Google Scholar 

  11. R. Froese, “Asymptotic Distribution of Resonances in One Dimension,” J. Differential Equations 137(2), 251–272 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  12. G. S. Guseinov, “Determination of an Infinite Jacobi Matrix from the Scattering Data,” Dokl. Akad. Nauk SSSR 227(6), 1289–1292 (1976) [Soviet Math. Dokl. 17, 596–600 (1976)].

    MathSciNet  Google Scholar 

  13. G. S. Guseinov, “The Inverse Problem of Scattering Theory for a Second-Order Difference Equation on the Whole Real Line,” Dokl. Akad. Nauk SSSR 230(6), 1045–1048 (1976) [Soviet Math. Dokl., 17, 1684–1688 (1976)].

    MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

  17. F. G. Maksudov, E. M. Bairamov, and R. U. Orudzheva, “An Inverse Scattering Problem for an Infinite Jacobi Matrix with Operator Elements,” Dokl. Akad. Nauk 323(3), 415–419 (1992) [Russ. Acad. Sci., Dokl., Math. 45 (2), 366–370 (1992) (1993)].

    MathSciNet  Google Scholar 

  18. V. Marchenko, The Sturm-Liouville Operator and Applications (Izdat. “Naukova Dumka”, Kiev, 1977; Birkhäuser, Basel, 1986).

    Google Scholar 

  19. V. Marchenko, Introduction to the Theory of Inverse Spectral Problems (Acta Publishing Company, Ukrain, 2005), [in Russian].

    Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. A. Melin, “Operator Methods for Inverse Scattering on the Real Line,” Comm. Partial Differential Equations 10(7), 677–766 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Novikov, S. Manakov, L. Pitaevski, and V. Zakharov, Theory of Solitons. The Inverse Scattering Method (Nauka, Moscow, 1980; Consultants Bureau [Plenum], New York-London, 1984).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, 72 (AMS, Providence, RI, 2000).

    MATH  Google Scholar 

  27. M. Toda, Theory of Nonlinear Lattices (2nd ed. Springer, Berlin-New York, 1989; Russian transl. of the 1st ed., Mir, Moscow, 1984).

    Book  MATH  Google Scholar 

  28. M. Toda, Theory of Nonlinear Waves and Solitons (Kluwer, Dordrecht, 1989).

    Google Scholar 

  29. M. Zworski, “Distribution of Poles for Scattering on the Real Line,” J. Funct. Anal. 73(2), 277–296 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Zworski, “A Remark on Isopolar Potentials,” SIAM J. Math. Anal. 32(6), 1324–1326 (2001).

    Article  MATH  MathSciNet  Google Scholar 

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  1. Mathematical Physics Department, Faculty of Physics, St. Petersburg State University, Ulianovskaya 2, St. Petersburg, 198904, Russia

    E. L. Korotyaev

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  1. E. L. Korotyaev
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Korotyaev, E.L. Inverse resonance scattering for Jacobi operators. Russ. J. Math. Phys. 18, 427–439 (2011). https://doi.org/10.1134/S1061920811040054

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  • DOI: https://doi.org/10.1134/S1061920811040054

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