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2011 | 9 | 3 | 657-672
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On the basis property of the root functions of differential operators with matrix coefficients

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We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.
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Bibliografia
  • [1] Dernek N., Veliev O.A., On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. Math., 2005, 145(1), 113–123 http://dx.doi.org/10.1007/BF02786687
  • [2] Djakov P., Mityagin B.S., Instability zones of periodic 1-dimensional Schrödinger and Dirac operators, Russian Math. Surveys, 2006, 61(4), 663–776 http://dx.doi.org/10.1070/RM2006v061n04ABEH004343
  • [3] Djakov P., Mityagin B.S., Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. (in press), DOI: 10.1007/s00208-010-0612-5
  • [4] Djakov P., Mityagin B.S., 1D Dirac operators with special periodic potentials, preprint available at http://arxiv.org/abs/1007.3234
  • [5] Dunford N., Schwartz J.T., Linear Operators. Part III: Spectral Operators, Mir, Moscow, 1974 (in Russian)
  • [6] Gohberg I.C., Krein M.G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Transl. Math. Monogr., 18, AMS, Providence, 1969
  • [7] Kato T., Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss., 132, Springer, Berlin, 1980
  • [8] Kerimov N.B., Mamedov Kh.R., On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes, 1998, 64(4), 483–487 http://dx.doi.org/10.1007/BF02314629
  • [9] Kesel’man G.M., On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat, 1964, 2, 82–93 (in Russian)
  • [10] Luzhina L.M., Regular spectral problems in a space of vector functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, 43(1), 31–35
  • [11] Makin A. S., On the convergence of expansions in root functions of a periodic boundary value problem, Dokl. Akad. Nauk, 2006, 406(4), 452–457 (in Russian)
  • [12] Mihaĭlov V.P., On Riesz bases in L 2[0,1], Dokl. Akad. Nauk SSSR 1962, 144, 981–984 (in Russian)
  • [13] Naimark M.A., Linear Differential Operators, Frederick Ungar Publishing Co., New York, 1967, 1968
  • [14] Shkalikov A.A., On the basis problem of the eigenfunctions of an ordinary differential operator, Russian Math. Surveys, 1979, 34(5), 249–250 http://dx.doi.org/10.1070/RM1979v034n05ABEH003901
  • [15] Shkalikov A.A., Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, 37(6), 12–21 (in Russian)
  • [16] Veliev O.A., Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Bound. Value Probl., 2008, ID 628973
  • [17] Veliev O.A., On the differential operators with periodic matrix coefficients, Abstr. Appl. Anal., 2009, ID 934905
  • [18] Veliev O.A., Shkalikov A.A., On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 2009, 85(5–6), 647–660
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-011-0015-1
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