Journal d’Analyse Mathématique

, Volume 73, Issue 1, pp 267–297 | Cite as

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

  • Fritz Gesztesy
  • Barry Simon


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 < ∞.


Inverse Problem Jacobi Matrix Spectral Measure Trace Formula Jacobi Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Z. S. Agranovich and V. A. Marchenko,The Inverse Problem of Scattering Theory, Gordon and Breach, New York, 1963.MATHGoogle Scholar
  2. [2]
    N. I. Akhiezer,The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965.Google Scholar
  3. [3]
    A. Antony and M. Krishna,Almost periodicity of some Jacobi matrices, Proc. Indian Acad. Sci.102 (1992), 175–188.MATHMathSciNetGoogle Scholar
  4. [4]
    A. Antony and M. Krishna,Inverse spectral theory for Jacobi matrices and their almost periodicity, Proc. Indian Acad. Sci.104 (1994), 777–818.MATHMathSciNetGoogle Scholar
  5. [5]
    G. A. Baker and P. Graves-Morris,Padé Approximants, Parts I and II, Addison-Wesley, Reading, 1981.Google Scholar
  6. [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.Google Scholar
  7. [7]
    Ju. M. Berezanskii,Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Math. Monographs17, Amer. Math. Soc., Providence, RI, 1968.Google Scholar
  8. [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.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    D. Boley and G. H. Golub,A survey of matrix inverse eigenvalue problems, Inverse Problems3 (1987), 595–622.MATHCrossRefMathSciNetGoogle Scholar
  10. [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.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Borg,Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math.78 (1946), 1–96.MATHCrossRefMathSciNetGoogle Scholar
  12. [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. [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.Google Scholar
  14. [14]
    K. M. Case,On discrete inverse scattering problems, II J. Math. Phys.14 (1973), 916–920.CrossRefMathSciNetGoogle Scholar
  15. [15]
    K. M. Case,Inverse problem in transport theory, Phys. Fluids16 (1973), 1607–1611.CrossRefMathSciNetGoogle Scholar
  16. [16]
    K. M. Case,The discrete inverse scattering problem in one dimension, J. Math. Phys.15 (1974), 143–146.CrossRefMathSciNetGoogle Scholar
  17. [17]
    K. M. Case,Scattering theory, orthogonal polynomials, and the transport equation, J. Math. Phys.15 (1974), 974–983.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    K. M. Case,Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys.15 (1974), 2166–2174.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    K. M. Case,Orthogonal polynomials, II, J. Math. Phys.16 (1975), 1435–1440.MATHCrossRefMathSciNetGoogle Scholar
  20. [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.CrossRefGoogle Scholar
  21. [21]
    K. M. Case and M. Kac,A discrete version of the inverse scattering problem, J. Math. Phys.14 (1973), 594–603.CrossRefMathSciNetGoogle Scholar
  22. [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.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    M. W. Davis,Some aspherical manifolds, Duke Math. J.55 (1987), 105–139.MATHCrossRefMathSciNetGoogle Scholar
  24. [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.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    P. Deift and E. Trubowitz,A continuum limit of matrix inverse problems, SIAM J. Math. Anal.12 (1981), 799–818.MATHCrossRefMathSciNetGoogle Scholar
  26. [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. [27]
    W. E. Ferguson,The construction of Jacobi and periodic Jacobi matrices with prescribed spectra, Math. Comp.35 (1980), 1203–1220.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    H. Flaschka,On the Toda lattice, II, Progr. Theoret. Phys.51 (1974), 703–716.CrossRefMathSciNetGoogle Scholar
  29. [29]
    D. Fried,The cohomology of an isospectral flow, Proc. Amer. Math. Soc.98 (1986), 363–368.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Fu and H. Hochstadt,Inverse theorems for Jacobi matrices, J. Math. Anal. Appl.47 (1974), 162–168.MATHCrossRefMathSciNetGoogle Scholar
  31. [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.MATHMathSciNetGoogle Scholar
  32. [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.MathSciNetGoogle Scholar
  33. [33]
    J. S. Geronimo and K. M. Case,Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys.20 (1979), 299–310.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    F. Gesztesy and H. Holden,Trace formulas and conservation laws for nonlinear evolution equations, Rev. Math. Phys.6 (1994), 51–95.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    F. Gesztesy and W. Renger,New classes of Toda soliton solutions, Comm. Math. Phys.184 (1997), 27–50.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    F. Gesztesy and B. Simon,The xi function, Acta Math.176 (1996), 49–71.MATHCrossRefMathSciNetGoogle Scholar
  37. [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.MATHCrossRefMathSciNetGoogle Scholar
  38. [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.MATHMathSciNetGoogle Scholar
  39. [39]
    F. Gesztesy and B. Simon,Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, preprint, 1997.Google Scholar
  40. [40]
    F. Gesztesy and G. Teschl,Commutation methods for Jacobi operators, J. Differential Equations128 (1996), 252–299.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    F. Gesztesy, M. Krishna and G. Teschl,On isospectral sets of Jacobi operators, Comm. Math. Phys.181 (1996), 631–645.MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    G. M. L. Gladwell,On isospectral spring-mass systems, Inverse Problems11 (1995), 591–602.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    W. B. Gragg and W. J. Harrod,The numerically stable reconstruction of Jacobi matrices from spectral data, Numer. Math.44 (1984), 317–335.MATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    L. J. Gray and D. G. Wilson,Construction of a Jacobi matrix from spectral data, Linear Algebra Appl.14 (1976), 131–134.MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    G.š. Guseinov,The determination of an infinite Jacobi matrix from the scattering data, Soviet Math. Dokl.17 (1976), 596–600.MATHGoogle Scholar
  46. [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.MATHGoogle Scholar
  47. [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. [48]
    O. Hald,Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl.14 (1976), 63–85.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    H. Hochstadt,On some inverse problems in matrix theory, Arch. Math.18 (1967), 201–207.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    H. Hochstadt,On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl.8 (1974), 435–446.MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    H. Hochstadt,On the construction of a Jacobi matrix from mixed given data, Linear Algebra Appl.28 (1979), 113–115.MATHCrossRefMathSciNetGoogle Scholar
  52. [52]
    H. Hochstadt and B. Lieberman,An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math.34 (1978), 676–680.CrossRefMathSciNetGoogle Scholar
  53. [53]
    M. Kac and P. van Moerbeke,On some periodic Toda lattices, Proc. Nat. Acad. Sci. U.S.A.72 (1975), 1627–1629.MATHCrossRefMathSciNetGoogle Scholar
  54. [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.MATHCrossRefMathSciNetGoogle Scholar
  55. [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.MATHCrossRefGoogle Scholar
  56. [56]
    H. J. Landau,The classical moment problem: Hilbertian proofs, J. Funct. Anal.38 (1980), 255–272.MATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    N. Levinson,The inverse Sturm-Liouville problem, Mat. Tidskr.B (1949), 25–30.MathSciNetGoogle Scholar
  58. [58]
    B. Levitan,On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl.68 (1968), 1–20.MathSciNetGoogle Scholar
  59. [59]
    B. Levitan,Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.MATHGoogle Scholar
  60. [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.CrossRefMathSciNetGoogle Scholar
  61. [61]
    B. Levitan and 1. Sargsjan,Sturm-Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.Google Scholar
  62. [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. [63]
    V. Marchenko,Sturm-Liouville Operators and Applications, BirkhÄuser, Basel, 1986.MATHGoogle Scholar
  64. [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.MATHCrossRefMathSciNetGoogle Scholar
  65. [65]
    P. van Moerbeke,The spectrum of Jacobi matrices, Invent. Math.37 (1976), 45–81.MATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    P. van Moerbeke and D. Mumford,The spectrum of difference operators and algebraic curves, Acta Math.143 (1979), 93–154.MATHCrossRefMathSciNetGoogle Scholar
  67. [67]
    J. Pöschel and E. Trubowitz,Inverse Scattering Theory, Academic Press, Boston, 1987.Google Scholar
  68. [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. [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.MathSciNetGoogle Scholar
  70. [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.Google Scholar
  71. [71]
    G. Szego,Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ.23, New York, 1939.Google Scholar
  72. [72]
    G. Teschl,Trace formulas and inverse spectral theory for Jacobi operators, preprint, 1996.Google Scholar
  73. [73]
    C. Tomei,The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J.51 (1984), 981–996.MATHCrossRefMathSciNetGoogle Scholar
  74. [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

Copyright information

© Hebrew University of Jerusalem 1997

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Division of Physics, Mathematics, and AstronomyCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations