On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev

Open Mathematics (2011)

  • Volume: 9, Issue: 3, page 657-672
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Oktay Veliev. "On the basis property of the root functions of differential operators with matrix coefficients." Open Mathematics 9.3 (2011): 657-672. <http://eudml.org/doc/269755>.

@article{OktayVeliev2011,
abstract = {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.},
author = {Oktay Veliev},
journal = {Open Mathematics},
keywords = {Differential operators; Eigenfunction expansion; differential operators; eigenfunction expansion; Sturm-Liouville operator; root functions; Riesz basis},
language = {eng},
number = {3},
pages = {657-672},
title = {On the basis property of the root functions of differential operators with matrix coefficients},
url = {http://eudml.org/doc/269755},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Oktay Veliev
TI - On the basis property of the root functions of differential operators with matrix coefficients
JO - Open Mathematics
PY - 2011
VL - 9
IS - 3
SP - 657
EP - 672
AB - 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.
LA - eng
KW - Differential operators; Eigenfunction expansion; differential operators; eigenfunction expansion; Sturm-Liouville operator; root functions; Riesz basis
UR - http://eudml.org/doc/269755
ER -

References

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  2. [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. [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 Zbl1242.34148
  4. [4] Djakov P., Mityagin B.S., 1D Dirac operators with special periodic potentials, preprint available at http://arxiv.org/abs/1007.3234 Zbl1256.47029
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  6. [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 Zbl0181.13504
  7. [7] Kato T., Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss., 132, Springer, Berlin, 1980 
  8. [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 Zbl0924.34072
  9. [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. [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. [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. [12] Mihaĭlov V.P., On Riesz bases in L 2[0,1], Dokl. Akad. Nauk SSSR 1962, 144, 981–984 (in Russian) 
  13. [13] Naimark M.A., Linear Differential Operators, Frederick Ungar Publishing Co., New York, 1967, 1968 Zbl0219.34001
  14. [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 Zbl0471.34014
  15. [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. [16] Veliev O.A., Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Bound. Value Probl., 2008, ID 628973 Zbl1158.34051
  17. [17] Veliev O.A., On the differential operators with periodic matrix coefficients, Abstr. Appl. Anal., 2009, ID 934905 Zbl1195.34134
  18. [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 Zbl1190.34111

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