Skip to main content
SpringerLink
Log in

Inverse Problem for a Stieltjes String Damped at an Interior Point

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper, we consider the spectral problem for a Stieltjes string with both ends fixed and with one-dimensional damping at an interior point. We solve an inverse problem of recovering parameters of the Stieltjes string using partial information on the string and partial information on the spectrum. First we prove uniqueness and give an algorithm of recovering the unknown parameters of the string by virtue of a finite number of distinct not pure imaginary complex eigenvalues and a certain even number of real eigenvalues. For the case where only complex eigenvalues are used we solve the existence problem, i.e. propose conditions on a set of complex numbers necessary and sufficient for these numbers to be a subset of the spectrum of the problem.

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

Similar content being viewed by others

References

  1. Arov, D.Z.: Realization of canonical system with a dissipative boundary condition at one end of the segment in terms of the coefficient of dynamical compliance. Z̆ Sibirsk. Mat. 16, 440–463 (1975)

    MathSciNet  Google Scholar 

  2. Boyko, O., Pivovarchik, V.: Inverse problem for Stieltjes string damped at one end. Methods Funct. Anal. Topology 14, 10–19 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Boyko, O., Pivovarchik, V.: The inverse three-spectral problem for a Stieltjes string and the inverse problem with one-dimensional damping. Inverse Probl. 24, 13 (2008)

    Article  MathSciNet  Google Scholar 

  4. Cox, S.J., Embree, M., Hokanson, J.M.: One can hear the composition of a string: experiments with an inverse eigenvalue problem. SIAM Rev. 54, 157–178 (2012)

    Article  MathSciNet  Google Scholar 

  5. Dudko, A., Pivovarchik, V.: Three spectra problem for Stieltjes string equation and Neumann conditions. Proc. Int. Geom. Cent. 12, 41–55 (2019)

    Article  MathSciNet  Google Scholar 

  6. Eckhardt, J., Kostenko, A.: The classical moment problem and generalized indefinite strings. Integr. Equ. Oper. Theory 90, 30 (2018)

    Article  MathSciNet  Google Scholar 

  7. Eckhardt, J., Kostenko, A.: The inverse spectral problem for indefinite strings. Invent. Math. 204, 939–977 (2016)

    Article  MathSciNet  Google Scholar 

  8. Filimonov, A.M., Kurchanov, P.F., Myshkis, A.D.: Some unexpected results in the classical problem of vibrations of the string with \(n\) beads when \(n\) is large. C.R. Acad. Sci. Paris Sér. I Math. 313, 961–965 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Filimonov, A.M., Myshkis, A.D.: On properties of large wave effect in classical problem of bead string vibration. J. Differ. Equ. Appl. 10, 1171–1175 (2004)

    Article  MathSciNet  Google Scholar 

  10. Gantmakher, F.R., Krein, M.G.: Oscillating Matrices and Kernels and Vibrations of Mechanical Systems, Berlin: German Transl. Akademie Verlag; 1960. GITTL, Moscow-Leningrad. Russian (1950)

  11. Gesztesy, F., Simon, B.: \(m\)-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73, 267–297 (1991)

    Article  MathSciNet  Google Scholar 

  12. Gladwell, G.: Inverse Problems in Vibration. Kluwer Academic Publishers, Dordrecht (2004)

    MATH  Google Scholar 

  13. Gladwell, G.: Matrix Inverse Eigenvalue Problems, Dynamical Inverse Problems: Theory and Applications. CISM Courses and Lectures, vol. 529. Springer, Vienna (2011)

    Book  Google Scholar 

  14. Horn, A.R., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  15. Kac, I.S., Krein, M.G.: \(R\)-functions-analytic functions mapping the upper half-plane into itself. Am. Math. Soc. Transl. 103, 1–18 (1974)

    Article  Google Scholar 

  16. Krein, M.G., Nudelman, A.A.: On direct and inverse problems for frequencies of boundary dissipation of inhomogeneous string. Doklady AN SSSR. 247, 1046–1049 (1979)

    Google Scholar 

  17. Krein, M.G., Nudelman, A.A.: Some spectral properties of a nonhomogeneous string with a dissipative boundary condition. J. Oper. Theory 22, 369–395 (1989)

    MathSciNet  Google Scholar 

  18. Krein, M.G., Nudelman, A.A.: An interpolation problem in the class of Stieltjes functions and its connection with other problems. Integr. Equ. Oper. Theory 30, 251–278 (1998)

    Article  MathSciNet  Google Scholar 

  19. Marchenko, V.A.: Introduction to The Theory of Inverse Problems of Spectral Analysis, Kharkov: Acta. Russian (2005)

  20. Martynuk, O., Pivovarchik, V., Tretter, C.: Inverse problem for a damped Stieltjes string from parts of spectra. Appl. Anal. 94, 2605–2619 (2015)

    Article  MathSciNet  Google Scholar 

  21. Möller, M., Pivovarchik, V.: Spectral theory of operator pencils, Hermite-Biehler functions, and their applications. In: Operator Theory: Advances and Applications, vol. 246, Birkhäuser/Springer, Cham (2015)

  22. Stieltjes, T.-L.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8, J1–J122 (1894)

    MathSciNet  MATH  Google Scholar 

  23. Stieltjes, T.-L.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 9, A5–A47 (1895)

    MathSciNet  MATH  Google Scholar 

  24. Veselić, K.: On linear vibrational systems with one dimensional damping I. Appl. Anal. 29, 1–18 (1988)

    Article  MathSciNet  Google Scholar 

  25. Veselić, K.: On linear vibrational systems with one dimensional damping II. Integral Equ. Oper. Theory 13, 883–897 (1990)

    Article  MathSciNet  Google Scholar 

  26. Yang, L., Guo, Y., Wei, G.: Inverse eigenvalue problems for a damped Stieltjes string with mixed data. Linear Algebra Appl. 601, 55–71 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their insightful suggestions and comments which improved and strengthened the presentation of this manuscript. The research was supported in part by the National Natural Science Foundation of China (11971284, 11601299), and the Fundamental Research Funds for the Central Universities (GK 201903002, GK 201903010, 2019TS086).

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Lu Yang, Yongxia Guo & Guangsheng Wei

Authors
  1. Lu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongxia Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guangsheng Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangsheng Wei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, L., Guo, Y. & Wei, G. Inverse Problem for a Stieltjes String Damped at an Interior Point. Integr. Equ. Oper. Theory 92, 30 (2020). https://doi.org/10.1007/s00020-020-02587-4

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-020-02587-4

Keywords

Mathematics Subject Classification

Search

Navigation