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
Z. S. Agranovich and V. A. Marchenko,The Inverse Problem of Scattering Theory, Gordon and Breach, New York, 1963.
N. I. Akhiezer,The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965.
A. Antony and M. Krishna,Almost periodicity of some Jacobi matrices, Proc. Indian Acad. Sci.102 (1992), 175–188.
A. Antony and M. Krishna,Inverse spectral theory for Jacobi matrices and their almost periodicity, Proc. Indian Acad. Sci.104 (1994), 777–818.
G. A. Baker and P. Graves-Morris,Padé Approximants, Parts I and II, Addison-Wesley, Reading, 1981.
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.
Ju. M. Berezanskii,Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Math. Monographs17, Amer. Math. Soc., Providence, RI, 1968.
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.
D. Boley and G. H. Golub,A survey of matrix inverse eigenvalue problems, Inverse Problems3 (1987), 595–622.
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.
G. Borg,Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math.78 (1946), 1–96.
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.
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.
K. M. Case,On discrete inverse scattering problems, II J. Math. Phys.14 (1973), 916–920.
K. M. Case,Inverse problem in transport theory, Phys. Fluids16 (1973), 1607–1611.
K. M. Case,The discrete inverse scattering problem in one dimension, J. Math. Phys.15 (1974), 143–146.
K. M. Case,Scattering theory, orthogonal polynomials, and the transport equation, J. Math. Phys.15 (1974), 974–983.
K. M. Case,Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys.15 (1974), 2166–2174.
K. M. Case,Orthogonal polynomials, II, J. Math. Phys.16 (1975), 1435–1440.
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.
K. M. Case and M. Kac,A discrete version of the inverse scattering problem, J. Math. Phys.14 (1973), 594–603.
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.
M. W. Davis,Some aspherical manifolds, Duke Math. J.55 (1987), 105–139.
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.
P. Deift and E. Trubowitz,A continuum limit of matrix inverse problems, SIAM J. Math. Anal.12 (1981), 799–818.
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.
W. E. Ferguson,The construction of Jacobi and periodic Jacobi matrices with prescribed spectra, Math. Comp.35 (1980), 1203–1220.
H. Flaschka,On the Toda lattice, II, Progr. Theoret. Phys.51 (1974), 703–716.
D. Fried,The cohomology of an isospectral flow, Proc. Amer. Math. Soc.98 (1986), 363–368.
L. Fu and H. Hochstadt,Inverse theorems for Jacobi matrices, J. Math. Anal. Appl.47 (1974), 162–168.
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.
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.
J. S. Geronimo and K. M. Case,Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys.20 (1979), 299–310.
F. Gesztesy and H. Holden,Trace formulas and conservation laws for nonlinear evolution equations, Rev. Math. Phys.6 (1994), 51–95.
F. Gesztesy and W. Renger,New classes of Toda soliton solutions, Comm. Math. Phys.184 (1997), 27–50.
F. Gesztesy and B. Simon,The xi function, Acta Math.176 (1996), 49–71.
F. Gesztesy and B. Simon,Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc.348 (1996), 349–373.
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.
F. Gesztesy and B. Simon,Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, preprint, 1997.
F. Gesztesy and G. Teschl,Commutation methods for Jacobi operators, J. Differential Equations128 (1996), 252–299.
F. Gesztesy, M. Krishna and G. Teschl,On isospectral sets of Jacobi operators, Comm. Math. Phys.181 (1996), 631–645.
G. M. L. Gladwell,On isospectral spring-mass systems, Inverse Problems11 (1995), 591–602.
W. B. Gragg and W. J. Harrod,The numerically stable reconstruction of Jacobi matrices from spectral data, Numer. Math.44 (1984), 317–335.
L. J. Gray and D. G. Wilson,Construction of a Jacobi matrix from spectral data, Linear Algebra Appl.14 (1976), 131–134.
G.š. Guseinov,The determination of an infinite Jacobi matrix from the scattering data, Soviet Math. Dokl.17 (1976), 596–600.
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.
G. Sh. Guseinov,Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function, Math. Notes23 (1978), 130–136.
O. Hald,Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl.14 (1976), 63–85.
H. Hochstadt,On some inverse problems in matrix theory, Arch. Math.18 (1967), 201–207.
H. Hochstadt,On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl.8 (1974), 435–446.
H. Hochstadt,On the construction of a Jacobi matrix from mixed given data, Linear Algebra Appl.28 (1979), 113–115.
H. Hochstadt and B. Lieberman,An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math.34 (1978), 676–680.
M. Kac and P. van Moerbeke,On some periodic Toda lattices, Proc. Nat. Acad. Sci. U.S.A.72 (1975), 1627–1629.
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.
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.
H. J. Landau,The classical moment problem: Hilbertian proofs, J. Funct. Anal.38 (1980), 255–272.
N. Levinson,The inverse Sturm-Liouville problem, Mat. Tidskr.B (1949), 25–30.
B. Levitan,On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl.68 (1968), 1–20.
B. Levitan,Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.
B. M. Levitan and M. G. Gasymov,Determination of a differential equation by two of its spectra, Russian Math. Surveys19:2 (1964), 1–63.
B. Levitan and 1. Sargsjan,Sturm-Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.
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.
V. Marchenko,Sturm-Liouville Operators and Applications, BirkhÄuser, Basel, 1986.
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.
P. van Moerbeke,The spectrum of Jacobi matrices, Invent. Math.37 (1976), 45–81.
P. van Moerbeke and D. Mumford,The spectrum of difference operators and algebraic curves, Acta Math.143 (1979), 93–154.
J. Pöschel and E. Trubowitz,Inverse Scattering Theory, Academic Press, Boston, 1987.
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.
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.
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.
G. Szego,Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ.23, New York, 1939.
G. Teschl,Trace formulas and inverse spectral theory for Jacobi operators, preprint, 1996.
C. Tomei,The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J.51 (1984), 981–996.
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.
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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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DOI: https://doi.org/10.1007/BF02788147