Skip to main content
SpringerLink
Log in

On the Properties of the Root Vector Function Systems of a \(2m \)th-Order Dirac Type Operator with an Integrable Potential

  • ORDINARY DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider a Dirac type operator with matrix coefficients. Estimates for the root vector functions are established, and criteria for the Bessel property and the unconditional basis property of the root vector function systems of this operator in the space \(L_{2}^{2m}(G) \), where \(G=(a,b)\subset \mathbb {R} \) is a finite interval, are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Il’in, V.A., On the unconditional basis property on a closed interval for systems of eigenfunctions and associated functions of a second-order differential operator, Dokl. Akad. Nauk SSSR, 1983, vol. 273, no. 5, pp. 1048–1053.

    MathSciNet  Google Scholar 

  2. Il’in, V.A., Necessary and sufficient conditions for basis property and equiconvergence with the trigonometric series of spectral expansions. I, Differ. Uravn., 1980, vol. 16, no. 5, pp. 771–794.

    Google Scholar 

  3. Budaev, V.D., Criteria for Bessel and Riesz basis properties of the systems of root functions of differential operators. I, Differ. Equations, 1991, vol. 27, no. 12, pp. 2033–2044.

    MATH  Google Scholar 

  4. Lomov, I.S., Estimates for eigenfunctions and associated functions of ordinary differential operators, Differ. Uravn., 1985, vol. 21, no. 5, pp. 903–906.

    MATH  Google Scholar 

  5. Kerimov, N.B., Some properties of eigenfunctions and associated functions of ordinary differential operators, Dokl. Akad. Nauk SSSR, 1986, vol. 201, no. 5, pp. 1054–1056.

    Google Scholar 

  6. Kurbanov, V.M., On the distribution of eigenvalues and the Bessel criterion for root functions of a differential operator. I, Differ. Equations, 2005, vol. 41, no. 4, pp. 489–505.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kritskov, L.V., Uniform estimate of the order of associated functions and distribution of eigenvalues of the one-dimensional Schrödinger operator, Differ. Equations, 1989, vol. 25, no. 7, pp. 784–791.

    MATH  Google Scholar 

  8. Tikhomirov, V.V., Sharp estimates for regular solutions of the one-dimensional Schrödinger equation with a spectral parameter, Dokl. Akad. Nauk SSSR, 1983, vol. 273, no. 4, pp. 807–810.

    MathSciNet  Google Scholar 

  9. Budaev, V.D., Criteria for Bessel and Riesz basis properties for the systems of root functions of differential operators. II, Differ. Equations, 1992, vol. 28, no. 1, pp. 21–30.

    MATH  Google Scholar 

  10. Lomov, I.S., Bessel’s inequality, Riesz theorem, and unconditional basis property for the root vectors of ordinary differential operators, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1992, no. 5, pp. 42–52.

  11. Kerimov, N.B., On the unconditional basis property of the system of eigenfunctions and associated functions of a fourth-order differential operator, Dokl. Akad. Nauk SSSR, 1986, vol. 286, no. 4, pp. 803–806.

    MathSciNet  Google Scholar 

  12. Kurbanov, V.M., On the eigenvalue distribution and a Bessel property criterion for root functions of a differential operator: II, Differ. Equations, 2005, vol. 41, no. 5, pp. 649–659.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kritskov, L.V. and Sarsenbi, A.M., Basicity in \(L_p \) of root functions for differential equations with involution, Electron. J. Differ. Equat., 2015, vol. 278, pp. 1–9.

    MathSciNet  MATH  Google Scholar 

  14. Kurbanov, V.M., On the Bessel property and unconditional basis property of the systems of root vector functions of the Dirac operator, Differ. Equations, 1996, vol. 32, no. 12, pp. 1601–1610.

    MathSciNet  MATH  Google Scholar 

  15. Kurbanov, V.M. and Gadzhieva, G.R., Bessel inequality and the basis property for a \( 2m\times 2m\) Dirac type system with an integrable potential, Differ. Equations, 2020, vol. 56, no. 5, pp. 573–584.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kurbanov, V.M. and Ismailova, A.I., Componentwise uniform equiconvergence of expansions in root vector functions of the Dirac operator with the trigonometric expansion, Differ. Equations, 2012, vol. 48, no. 5, pp. 655–669.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kurbanov, V.M. and Ismailova, A.I., Absolute and uniform convergence of expansions in root vector functions of the Dirac operator, Dokl. Ross. Akad. Nauk, 2012, vol. 446, no. 4, pp. 380–383.

    MathSciNet  MATH  Google Scholar 

  18. Kurbanov, V.M. and Ismailova, A.I., Riesz inequality for systems of root vector functions of the Dirac operator, Differ. Equations, 2012, vol. 48, no. 3, pp. 336–342.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kurbanov, V.M. and Ismailova, A.I., Two-sided estimates for root vector functions of the Dirac operator, Differ. Equations, 2012, vol. 48, no. 4, pp. 494–505.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kurbanov, V.M. and Abdullayeva, A.M., Bessel property and basicity of the system of root vector-functions of Dirac operator with summable coefficient, Oper. Matrices, 2018, vol. 12, no. 4, pp. 943–954.

    Article  MathSciNet  MATH  Google Scholar 

  21. Oridoroga, L.L. and Hassi, S., Completeness and Riesz basis property of systems of eigenfunctions and associated functions of Dirac-type operators with boundary conditions depending on the spectral parameter, Math. Notes, 2006, vol. 79, no. 4, pp. 589–593.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lunev, A.A. and Malamud, M.M., On the basis property of the system of Riesz root vectors for a \(2\times 2\) Dirac type system, Dokl. Ross. Akad. Nauk, 2014, vol. 458, no. 3, pp. 255–260.

    Google Scholar 

  23. Lunyov, A.A. and Malamud, M.M., On the Riesz basis property of the root vector system for Dirac-type systems, J. Math. Anal. Appl., 2016, vol. 441, no. 1, pp. 57–103.

    Article  MathSciNet  MATH  Google Scholar 

  24. Lunyov, A.A. and Malamud, M.M., On the completeness and Riesz basis property of root supspaces of boundary value problems for first order systems and applications, J. Spectr. Theory, 2015, vol. 5, pp. 17–70.

    Article  MathSciNet  MATH  Google Scholar 

  25. Lunev, A.A. and Malamud, M.M., On the characteristic determinants of boundary value problems for a Dirac type system, Zap. Nauchn. Semin. POMI, 2022, vol. 516, pp. 69–120.

    MathSciNet  Google Scholar 

  26. Savchuk, A.M. and Shkalikov, A.A., The Dirac operator with complex-valued summable potential, Math. Notes, 2014, vol. 96, no. 5, pp. 777–810.

    Article  MathSciNet  MATH  Google Scholar 

  27. Savchuk, A.M. and Sadovnichaya, I.V., The Riesz basis property with brackets for Dirac systems with summable potentials, J. Math. Sci., 2018, vol. 233, no. 4, pp. 514–540.

    Article  MathSciNet  MATH  Google Scholar 

  28. Trooshin, I. and Yamamota, M., Riesz basis of root vector of a nonsymmetric system of first-order ordinary differential operators and application to inverse eigenvalue problems, Appl. Anal., 2001, vol. 80, no. 1–2, pp. 19–51.

    MathSciNet  Google Scholar 

  29. Djakov, P. and Mityagin, B., Criteria for existence of Riesz basis consisting of root functions of Hill and 1D Dirac operators, J. Func. Anal., 2012, vol. 263, pp. 2300–2332.

    Article  MATH  Google Scholar 

  30. Djakov, P. and Mityagin, B., Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions, Indiana Univ. Math. J., 2012, vol. 61, no. 1, pp. 359–398.

    Article  MathSciNet  MATH  Google Scholar 

  31. Mykytnyk, Ya.V. and Puyda, D.V., Bari–Markus property of Dirac operators, Mat. Stud., 2013, vol. 40, no. 2, pp. 165–171.

    MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGMENTS

The author expresses deep gratitude to Prof. V.M. Kurbanov for his constant attention to the work.

Author information

Authors and Affiliations

  1. Azerbaijan State Pedagogical University, Baku, 370000, Azerbaijan

    E. C. Ibadov

Authors
  1. E. C. Ibadov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. C. Ibadov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibadov, E.C. On the Properties of the Root Vector Function Systems of a \(2m \)th-Order Dirac Type Operator with an Integrable Potential. Diff Equat 59, 1295–1314 (2023). https://doi.org/10.1134/S00122661230100014

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S00122661230100014

Search

Navigation