Skip to main content
SpringerLink
Log in

Asymptotics of Eigenvalues of Fourth Order Differential Operator with Alternating Weight Function

  • Published:
Moscow University Mathematics Bulletin Aims and scope

Abstract

We study a fourth-order differential operator with a sign-alternating weight function with separated boundary conditions. For large values of the spectral parameter the asymptotics of the solutions to the corresponding differential equations is derived. The study of boundary conditions makes it possible to obtain an equation for eigenvalues of the considered differential operator. The indicator diagram of this equation is studied. The asymptotics of eigenvalues in various sectors of the indicator diagram is 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.

Similar content being viewed by others

References

  1. V. A. Il’in, “Convergence of Expansions over Eigenfunctions at Points of Discontinuity of Differential Operator,” Matem. Zametki 22(5), 698 (1977).

    Google Scholar 

  2. V. D. Budaev, “Unconditional Basis Property on a Closed Interval for Systems of Eigenfunctions and Associated Functions,” Differ. Uravn. 23(6), 941 (1987).

    MATH  Google Scholar 

  3. H. P. W. Gottlieb, “Iso-Spectral Operators: Some Model Examples with Discontinuous Coefficients.” J. Math. Anal. and Appl. 132, 123 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. I. Mitrokhin, “Some Spectral Properties of Second Order Differential Operators with a Discontinuous Weight Function,” Doklady Russ. Akad. Nauk. 356(1), 13 (1997).

    MathSciNet  MATH  Google Scholar 

  5. A. P. Gurevich and A. P. Khromov, “Differential Operators of First and Second Orders with Alternating Weight Function,” Matem. Zametki 56(1), 3 (1994).

    Google Scholar 

  6. S. I. Mitrokhin, Spectral Theory of Operators: Smooth, Discontinuous, Summable Coefficient (INTUIT, Moscow, 2009) [in Russian].

    Google Scholar 

  7. S. I. Mitrokhin, “Spectral Properties of Differential Operator with Summable Potential and Smooth Weight Function,” Vestn. SamGU. Estestv. Ser., No. 8(1/67), 172 (2008).

    Google Scholar 

  8. S. I. Mitrokhin, “Spectral Properties of Differential Operators of Odd Order with Summable Potential,” Differ. Uravn. 47(12), 1808 (2011).

    MathSciNet  Google Scholar 

  9. M. A. Naimark, Linear Differential Operators (Harrap, L., Toronto, 1968; Nauka, Moscow, 1969).

    Google Scholar 

  10. B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory (Nauka, Moscow, 1970) [in Russian].

    MATH  Google Scholar 

  11. R. Bellman and K. L. Cooke, Differential-Difference Equations (Acad. Press, N.Y., L. 1963; Mir, Moscow, 1967).

    MATH  Google Scholar 

  12. V. A. Sadovnichii and V. A. Lyubishkin, “Some New Results of the Theory of Regularized Traces of Differential Operators,” Differ. Uravn. 18(1), 109 (1982).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Scientific Research Computing Center, Leninskie Gory, Moscow, 119991, Russia

    S. I. Mitrokhin

Authors
  1. S. I. Mitrokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. I. Mitrokhin.

Additional information

Original Russian Text © S.I. Mitrokhin, 2018, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2018, Vol. 73, No. 6, pp. 46–58.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mitrokhin, S.I. Asymptotics of Eigenvalues of Fourth Order Differential Operator with Alternating Weight Function. Moscow Univ. Math. Bull. 73, 254–265 (2018). https://doi.org/10.3103/S0027132218060074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0027132218060074

Search

Navigation