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On the Spectrum of Some Class of Jacobi Operators in a Krein Space

  • I. A. SheipakEmail author
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

A Jacobi matrix with exponential growth of its elements and a corresponding symmetric operator are considered.I t is proved that the eigenvalue problem of some self-adjoint extension of the operator in some Hilbert space is equivalent to the eigenvalue problem of a Sturm–Liouville operator with discrete self-similar weight.A n asymptotic formula for the eigenvalues distribution is obtained.T he case of an indefinite metric and self-adjoint extension of the operator in a Krein space is also considered.

Keywords

Jacobi matrix self-adjoint extension of the symmetric operators eigenvalues asymptotic self-similar function Krein space 

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References

  1. 1.
    E.A. Tur Eigenvalue asymptotic for one class of Jacobi matrix with limit point spectra // Matem. zametki, 2003, 74:3, 449-462 (in Russian); English transl.: Mathem. Notes, 2003, 74:3, 425–437.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.V. Kozhan Asymptotics of the eigenvalues of two-diagonal Jacobi matrices // Matem. zametki, 2005 77:2, p. 313–316 (in Russian); English transl.: Mathem. Notes, 2005 77:2, p. 283–287.MathSciNetCrossRefGoogle Scholar
  3. 3.
    F.R. Gantmakher, M.G. Krein Oscillation matrix and kernels and small vibrations of mechanical systems // GITTL, Moscow, Leningrad, 1950 (in Russian).Google Scholar
  4. 4.
    A.A. Vladimirov, I.A. Sheipak, Self-similar functions in space L2 [0,1] and Sturm-Liouville problem with singular weight, Matem. sbornik, 2006, 197:11, 13–30 (in Russian); English transl.: Sbornik: Mathematics, 2006, 197:11, 1569–1586.MathSciNetCrossRefGoogle Scholar
  5. 5.
    I.A. Sheipak, On the construction and some properties of self-similar functions in the spaces Lp [0,1], 2007, Matem. zametki, 81:6, 924–938 (in Russian); English transl.: Mathem. Notes, 2007, 81:5–6, 827–839.MathSciNetCrossRefGoogle Scholar
  6. 6.
    I.A. Sheipak, Singular Points of a Self-Similar Function of Spectral Order Zero: Self-Similar Stieltjes String, Matem. zametki, 2010, 88:2, 303–316 (in Russian); English transl.: Mathem. Notes, 2010 88:2, 275–286.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A.A. Vladimirov, I.A. Sheipak, Indefinite Sturm-Liouville problem for some classes of self-similar singular weights, Trudy MIRAN, 2006, 255, 88–98 (in Russian); English transl.: Proceedings of the Steklov Institute of Mathematics, 2006, 255, 1–10.MathSciNetGoogle Scholar
  8. 8.
    A.A. Vladimirov, I.A. Sheipak, Eigenvalue asymptotics for Sturm–Liouville problem with discrete self-similar weight // Matem. zametki, 2010, 88:5, 662–672 (in Russian); English transl.: Mathem. Notes, 2010 88:5, 3–12.MathSciNetCrossRefGoogle Scholar
  9. 9.
    N.I. Ahiezer Classic moment problems and some connected calculus problems // Moscow, Fizmatgiz., 1961 (in Russian).Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Moscow Lomonosov State UniversityMoscowRussia

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