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m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

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Abstract

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH). We prove an extension of the theorem of Hochstadt (who proved the result in casen = N) thatn eigenvalues of anN × N Jacobi matrixH can replace the firstn matrix elements in determiningH uniquely. We completely solve the inverse problem for (δ n , (H-z)-1 δ n ) in the caseN < ∞.

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References

  1. Z. S. Agranovich and V. A. Marchenko,The Inverse Problem of Scattering Theory, Gordon and Breach, New York, 1963.

    MATH  Google Scholar 

  2. N. I. Akhiezer,The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965.

    Google Scholar 

  3. A. Antony and M. Krishna,Almost periodicity of some Jacobi matrices, Proc. Indian Acad. Sci.102 (1992), 175–188.

    MATH  MathSciNet  Google Scholar 

  4. A. Antony and M. Krishna,Inverse spectral theory for Jacobi matrices and their almost periodicity, Proc. Indian Acad. Sci.104 (1994), 777–818.

    MATH  MathSciNet  Google Scholar 

  5. G. A. Baker and P. Graves-Morris,Padé Approximants, Parts I and II, Addison-Wesley, Reading, 1981.

    Google Scholar 

  6. D. BÄttig, B. Grébert, J.-C. Guillot and T. Kappeler,Fibration of the phase space of the periodic Toda lattice, J. Math. Pures Appl., to appear.

  7. Ju. M. Berezanskii,Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Math. Monographs17, Amer. Math. Soc., Providence, RI, 1968.

    Google Scholar 

  8. A. M. Bloch, H. Flaschka and T. Ratiu,A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra, Duke Math. J.61 (1990), 41–65.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Boley and G. H. Golub,A survey of matrix inverse eigenvalue problems, Inverse Problems3 (1987), 595–622.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. de Boor and G.H. Golub,The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra Appl.21 (1978), 245–260.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Borg,Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math.78 (1946), 1–96.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Borg,Uniqueness theorems in the spectral theory of y n + (δ-q(x))y = 0, Proc. 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, pp. 276–287.

    Google Scholar 

  13. W. Bulla, F. Gesztesy, H. Holden and G. Teschl,Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies, Mem. Amer. Math. Soc., to appear.

  14. K. M. Case,On discrete inverse scattering problems, II J. Math. Phys.14 (1973), 916–920.

    Article  MathSciNet  Google Scholar 

  15. K. M. Case,Inverse problem in transport theory, Phys. Fluids16 (1973), 1607–1611.

    Article  MathSciNet  Google Scholar 

  16. K. M. Case,The discrete inverse scattering problem in one dimension, J. Math. Phys.15 (1974), 143–146.

    Article  MathSciNet  Google Scholar 

  17. K. M. Case,Scattering theory, orthogonal polynomials, and the transport equation, J. Math. Phys.15 (1974), 974–983.

    Article  MATH  MathSciNet  Google Scholar 

  18. K. M. Case,Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys.15 (1974), 2166–2174.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. M. Case,Orthogonal polynomials, II, J. Math. Phys.16 (1975), 1435–1440.

    Article  MATH  MathSciNet  Google Scholar 

  20. K. M. Case and S. C. Chui,The discrete version of the Marchenko equations in the inverse scattering problem, J. Math. Phys.14 (1973), 1643–1647.

    Article  Google Scholar 

  21. K. M. Case and M. Kac,A discrete version of the inverse scattering problem, J. Math. Phys.14 (1973), 594–603.

    Article  MathSciNet  Google Scholar 

  22. E. Date and S. Tanaka,Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice, Progr. Theoret. Phys.55 (1976), 457–465.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. W. Davis,Some aspherical manifolds, Duke Math. J.55 (1987), 105–139.

    Article  MATH  MathSciNet  Google Scholar 

  24. P. Deift and T. Nanda,On the determination of a tridiagonal matrix from its spectrum and a submatrix. Linear Algebra Appl.60 (1984), 43–55.

    Article  MATH  MathSciNet  Google Scholar 

  25. P. Deift and E. Trubowitz,A continuum limit of matrix inverse problems, SIAM J. Math. Anal.12 (1981), 799–818.

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Deift, L.-C. Li and C. Tomei,Symplectic aspects of some eigenvalue algorithms, inImportant Developments in Soliton Theory (A. S. Fokas and V. E. Zakharov, eds.), Springer, Berlin, 1993, pp. 511–536.

    Google Scholar 

  27. W. E. Ferguson,The construction of Jacobi and periodic Jacobi matrices with prescribed spectra, Math. Comp.35 (1980), 1203–1220.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Flaschka,On the Toda lattice, II, Progr. Theoret. Phys.51 (1974), 703–716.

    Article  MathSciNet  Google Scholar 

  29. D. Fried,The cohomology of an isospectral flow, Proc. Amer. Math. Soc.98 (1986), 363–368.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Fu and H. Hochstadt,Inverse theorems for Jacobi matrices, J. Math. Anal. Appl.47 (1974), 162–168.

    Article  MATH  MathSciNet  Google Scholar 

  31. M. G. Gasymov and G. Sh. Guseinov,On inverse problems of spectral analysis for infinite Jacobi matrices in the limit-circle case, Soviet Math. Dokl.40 (1990), 627–630.

    MATH  MathSciNet  Google Scholar 

  32. I. M. Gel’fand and B. M. Levitan,On the determination of a differential equation from its special function, Izv. Akad. Nauk SSR, Ser. Mat.15 (1951), 309–360 (Russian); English transl. in Amer. Math. Soc. Transl. Ser. 21 (1955), 253–304.

    MathSciNet  Google Scholar 

  33. J. S. Geronimo and K. M. Case,Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys.20 (1979), 299–310.

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Gesztesy and H. Holden,Trace formulas and conservation laws for nonlinear evolution equations, Rev. Math. Phys.6 (1994), 51–95.

    Article  MATH  MathSciNet  Google Scholar 

  35. F. Gesztesy and W. Renger,New classes of Toda soliton solutions, Comm. Math. Phys.184 (1997), 27–50.

    Article  MATH  MathSciNet  Google Scholar 

  36. F. Gesztesy and B. Simon,The xi function, Acta Math.176 (1996), 49–71.

    Article  MATH  MathSciNet  Google Scholar 

  37. F. Gesztesy and B. Simon,Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc.348 (1996), 349–373.

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Gesztesy and B. Simon,Inverse spectral analysis with partial information on the potential, I. The case of an a.c. component in the spectrum, Helv. Phys. Acta70 (1997), 66–71.

    MATH  MathSciNet  Google Scholar 

  39. F. Gesztesy and B. Simon,Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, preprint, 1997.

  40. F. Gesztesy and G. Teschl,Commutation methods for Jacobi operators, J. Differential Equations128 (1996), 252–299.

    Article  MATH  MathSciNet  Google Scholar 

  41. F. Gesztesy, M. Krishna and G. Teschl,On isospectral sets of Jacobi operators, Comm. Math. Phys.181 (1996), 631–645.

    Article  MATH  MathSciNet  Google Scholar 

  42. G. M. L. Gladwell,On isospectral spring-mass systems, Inverse Problems11 (1995), 591–602.

    Article  MATH  MathSciNet  Google Scholar 

  43. W. B. Gragg and W. J. Harrod,The numerically stable reconstruction of Jacobi matrices from spectral data, Numer. Math.44 (1984), 317–335.

    Article  MATH  MathSciNet  Google Scholar 

  44. L. J. Gray and D. G. Wilson,Construction of a Jacobi matrix from spectral data, Linear Algebra Appl.14 (1976), 131–134.

    Article  MATH  MathSciNet  Google Scholar 

  45. G.š. Guseinov,The determination of an infinite Jacobi matrix from the scattering data, Soviet Math. Dokl.17 (1976), 596–600.

    MATH  Google Scholar 

  46. G.S. Guseinov,The inverse problem of scattering theory for a second-order difference equation of the whole axis, Soviet. Math. Dokl.17 (1976), 1684–1688.

    MATH  Google Scholar 

  47. G. Sh. Guseinov,Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function, Math. Notes23 (1978), 130–136.

    Google Scholar 

  48. O. Hald,Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl.14 (1976), 63–85.

    Article  MATH  MathSciNet  Google Scholar 

  49. H. Hochstadt,On some inverse problems in matrix theory, Arch. Math.18 (1967), 201–207.

    Article  MATH  MathSciNet  Google Scholar 

  50. H. Hochstadt,On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl.8 (1974), 435–446.

    Article  MATH  MathSciNet  Google Scholar 

  51. H. Hochstadt,On the construction of a Jacobi matrix from mixed given data, Linear Algebra Appl.28 (1979), 113–115.

    Article  MATH  MathSciNet  Google Scholar 

  52. H. Hochstadt and B. Lieberman,An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math.34 (1978), 676–680.

    Article  MathSciNet  Google Scholar 

  53. M. Kac and P. van Moerbeke,On some periodic Toda lattices, Proc. Nat. Acad. Sci. U.S.A.72 (1975), 1627–1629.

    Article  MATH  MathSciNet  Google Scholar 

  54. M. Kac and P. van Moerbeke,A complete solution of the periodic Toda problem, Proc. Nat. Acad. Sci. U.S.A.72 (1975), 2879–2880.

    Article  MATH  MathSciNet  Google Scholar 

  55. M. Kac and P. van Moerbeke,On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math.16 (1975), 160–169.

    Article  MATH  Google Scholar 

  56. H. J. Landau,The classical moment problem: Hilbertian proofs, J. Funct. Anal.38 (1980), 255–272.

    Article  MATH  MathSciNet  Google Scholar 

  57. N. Levinson,The inverse Sturm-Liouville problem, Mat. Tidskr.B (1949), 25–30.

    MathSciNet  Google Scholar 

  58. B. Levitan,On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl.68 (1968), 1–20.

    MathSciNet  Google Scholar 

  59. B. Levitan,Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.

    MATH  Google Scholar 

  60. B. M. Levitan and M. G. Gasymov,Determination of a differential equation by two of its spectra, Russian Math. Surveys19:2 (1964), 1–63.

    Article  MathSciNet  Google Scholar 

  61. B. Levitan and 1. Sargsjan,Sturm-Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.

    Google Scholar 

  62. V. A. Marchenko,Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy Moskov. Mat. Obšč.1 (1952), 327–420 (Russian); English transl. in Amer. Math. Soc. Transl. (2)101 (1973), 1–104.

    Google Scholar 

  63. V. Marchenko,Sturm-Liouville Operators and Applications, BirkhÄuser, Basel, 1986.

    MATH  Google Scholar 

  64. D. Masson and J. Repka,Spectral theory of Jacobi matrices in ℓ 2 (ℤ) and the SU[1, 1] Lie algebra, SIAM J. Math. Anal.22 (1991), 1131–1145.

    Article  MATH  MathSciNet  Google Scholar 

  65. P. van Moerbeke,The spectrum of Jacobi matrices, Invent. Math.37 (1976), 45–81.

    Article  MATH  MathSciNet  Google Scholar 

  66. P. van Moerbeke and D. Mumford,The spectrum of difference operators and algebraic curves, Acta Math.143 (1979), 93–154.

    Article  MATH  MathSciNet  Google Scholar 

  67. J. Pöschel and E. Trubowitz,Inverse Scattering Theory, Academic Press, Boston, 1987.

    Google Scholar 

  68. B. Simon,Spectral analysis of rank one perturbations and applications, CRM Lecture Notes Vol. 8 (J. Feldman, R. Froese and L. Rosen, eds.), Amer. Math. Soc., Providence, RI, 1995, pp. 109–149.

    Google Scholar 

  69. M. L. Sodin and P. M. Yuditskii,Infinite-zone Jacobi matrices with pseudo-extendible Weyl functions and homogeneous spectrum, Russian Acad. Sci. Dokl. Math.49 (1994), 364–368.

    MathSciNet  Google Scholar 

  70. M. L. Sodin and P. M. Yuditskil,Infinite-dimensional Jacobi inversion problem, almost-periodic Jacobi matrices with homogeneous spectrum, and Hardy classes of character-automorphic functions, preprint, 1994.

  71. G. Szego,Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ.23, New York, 1939.

  72. G. Teschl,Trace formulas and inverse spectral theory for Jacobi operators, preprint, 1996.

  73. C. Tomei,The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J.51 (1984), 981–996.

    Article  MATH  MathSciNet  Google Scholar 

  74. B. N. Zakhar’iev, V. N. Mel’nikov, B. V. Rudyak and A. A. Suz’ko,Inverse scattering problem (finite-difference approach), Soviet J. Part. Nucl.8 (1977), 120–137.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    Fritz Gesztesy

  2. Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 91125, Pasadena, CA, USA

    Barry Simon

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  1. Fritz Gesztesy
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Correspondence to Fritz Gesztesy.

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This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.

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Gesztesy, F., Simon, B. m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73, 267–297 (1997). https://doi.org/10.1007/BF02788147

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