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On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition

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Abstract

We consider the spectral problem for the Schrödinger operator with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have multiple eigenvalues and a system of eigenfunctions forming a Riesz basis in L 2(0, 1). We show that the basis property of systems of root functions of the problem can change under arbitrarily small changes in the kernel of the integral perturbation.

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References

  1. Il’in, V.A., On the Relationship between the Form of the Boundary Conditions and the Basis Property and Property of Equiconvergence with the Trigonometric Series of Expansions in Root Functions of a Nonself-Adjoint Differential Operator, Differ. Uravn., 1994, vol. 30, no. 9, pp. 1516–1529.

    Google Scholar 

  2. Makin, A.S., On a Nonlocal Perturbation of a Periodic Eigenvalue Problem, Differ. Uravn., 2006, vol. 42, no. 4, pp. 560–562.

    MathSciNet  Google Scholar 

  3. Shkalikov, A.A., Basis Property of Eigenfunctions of Ordinary Differential Operators with Integral Boundary Conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, no. 6, pp. 12–21.

  4. Makin, A.S., On the Convergence of Expansions in Root Functions of a Periodic Boundary Value Problem, Dokl. Akad. Nauk, 2006, vol. 406, no. 4, pp. 452–457.

    MathSciNet  Google Scholar 

  5. Veliev, O.A. and Shkalikov, A.A., On the Riesz Basis Property of Eigen- and Associated Functions of Periodic and Anti-Periodic Sturm-Liouville Problems, Mat. Zametki, 2009, vol. 85, no. 5, pp. 671–686.

    MathSciNet  Google Scholar 

  6. Tkachenko, V.A., Spectral Analysis of the Nonself-Adjoint Hill Operator, Dokl. Akad. Nauk, 1992, vol. 322, no. 2, pp. 248–252.

    Google Scholar 

  7. Dezin, A.A., Obshchie voprosy teorii granichnykh zadach (General Questions of the Theory of Boundary Value Problems), Moscow: Nauka, 1980.

    MATH  Google Scholar 

  8. Otelbaev, M. and Kokebaev, B.K., On a Class of Ill-Posed Boundary Value Problems, Uspekhi Mat. Nauk, 1982, vol. 37, no. 4, pp. 116–227.

    Google Scholar 

  9. Lomov, I.S., The Basis Property of Root Vectors of Loaded Second-Order Differential Operators on an Interval, Differ. Uravn., 1991, vol. 27, no. 1, pp. 80–93.

    MathSciNet  Google Scholar 

  10. Lomov, I.S., A Theorem on the Unconditional Basis Property of Root Vectors of Second-Order Weighted Differential Operators, Differ. Uravn., 1991, vol. 27, no. 9, pp. 1550–1563.

    MathSciNet  MATH  Google Scholar 

  11. Funktsional’nyi analiz (Functional Analysis), Krein, S.G., Ed., Moscow: Nauka, 1972.

    Google Scholar 

  12. Bari, N.K., Biorthogonal Systems and Bases in Hilbert Space, Uch. Zap. Moskov. Gos. Univ., 1951, vol. 4, no. 148, pp. 69–107.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Mechanics, and Computer Science, National Academy of Sciences, Almaty, Kazakhstan International Kazakhstan-Turkey University, Shymkent, Kazakhstan

    M. A. Sadybekov & N. S. Imanbaev

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  1. M. A. Sadybekov
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  2. N. S. Imanbaev
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Original Russian Text © M.A. Sadybekov, N.S. Imanbaev, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 6, pp. 889–893.

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Sadybekov, M.A., Imanbaev, N.S. On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition. Diff Equat 48, 896–900 (2012). https://doi.org/10.1134/S0012266112060146

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  • DOI: https://doi.org/10.1134/S0012266112060146

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