Skip to main content
Log in

On recovering differential systems on a finite interval from spectra

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

Inverse spectral problems are studied for non-self-adjoint systems of ordinary differential equations on a finite interval. We establish properties of spectral characteristics and provide a procedure for constructing the solution of the inverse problem of recovering the coefficients of differential systems from given spectra.

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

Access this article

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. Yurko, Method of Spectral Mappings in the Inverse Problem Theory (VSP, Utrecht, 2002), Inverse Ill-Posed Probl. Ser.

    MATH  Google Scholar 

  2. V. A. Yurko, Inverse Spectral Problems for Differential Operators and Their Applications (Gordon and Breach, Amsterdam, 2000).

    MATH  Google Scholar 

  3. V. A. Yurko, Inverse Spectral Problems and Their Applications (Saratov Pedagog. Inst., Saratov, 2001) [in Russian].

    Google Scholar 

  4. B. M. Levitan and I. S. Sargsyan, Sturm-Liouville and Dirac Operators (Nauka, Moscow, 1988; Kluwer, Dordrecht, 1991).

    MATH  Google Scholar 

  5. V. A. Marchenko, Sturm-Liouville Operators and Their Applications (Naukova Dumka, Kiev, 1977; Birkhäuser, Basel, 1986).

    MATH  Google Scholar 

  6. B. M. Levitan, Inverse Sturm-Liouville Problems (Nauka, Moscow, 1984; VNU Sci. Press, Utrecht, 1987).

    MATH  Google Scholar 

  7. G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and Their Applications (Nova Sci. Publ., Huntington, NY, 2001).

    MATH  Google Scholar 

  8. A. B. Shabat, “An Inverse Scattering Problem,” Diff. Uravn. 15(10), 1824–1834 (1979) [Diff. Eqns. 15, 1299–1307 (1979)].

    MathSciNet  Google Scholar 

  9. V. A. Yurko, “An Inverse Problem for Systems of Differential Equations with Nonlinear Dependence on the Spectral Parameter,” Diff. Uravn. 33(3), 390–395 (1997) [Diff. Eqns. 33, 388–394 (1997)].

    MathSciNet  Google Scholar 

  10. M. M. Malamud, “Uniqueness Questions in Inverse Problems for Systems of Differential Equations on a Finite Interval,” Tr. Mosk. Mat. O-va. 60, 199–258 (1999) [Trans. Moscow Math. Soc. 60, 173–224 (1999)].

    MathSciNet  Google Scholar 

  11. L. A. Sakhnovich, Spectral Theory of Canonical Differential Systems. Method of Operator Identities (Birkhäuser, Basel, 1999), Operator Theory: Adv. Appl. 107.

    MATH  Google Scholar 

  12. R. Beals and R. R. Coifman, “Scattering and Inverse Scattering for First Order Systems,” Commun. Pure Appl. Math. 37, 39–90 (1984).

    MathSciNet  Google Scholar 

  13. V. A. Yurko, “An Inverse Spectral Problem for Singular Non-self-adjoint Differential Systems,” Mat. Sb. 195(12), 123–156 (2004) [Sb. Math. 195, 1823–1854 (2004)].

    MathSciNet  Google Scholar 

  14. V. A. Yurko, “An Inverse Problem for Differential Systems on a Finite Interval in the Case of Multiple Roots of the Characteristic Polynomial,” Diff. Uravn. 41(6), 781–786 (2005) [Diff. Eqns. 41, 818–823 (2005)].

    MathSciNet  Google Scholar 

  15. Ya. D. Tamarkin, Some General Problems of the Theory of Ordinary Linear Differential Equations and Expansion of an Arbitrary Function in Series of Fundamental Functions (Petrograd, 1917) [in Russian]; abridged Engl. transl. in Math. Z. 27, 1–54 (1928).

  16. M. L. Rasulov, Contour Integral Method (Nauka, Moscow, 1964) [in Russian].

    Google Scholar 

  17. A. I. Vagabov, “On Sharpening an Asymptotic Theorem of Tamarkin,” Diff. Uravn. 29(1), 41–49 (1993) [Diff. Eqns. 29, 33–41 (1993)].

    MathSciNet  Google Scholar 

  18. V. S. Rykhlov, “Asymptotical Formulas for Solutions of Linear Differential Systems of the First Order,” Results Math. 36(3–4), 342–353 (1999).

    MathSciNet  Google Scholar 

  19. R. Bellmann and K. Cooke, Differential-Difference Equations (Academic, New York, 1963).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 273–287.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yurko, V.A. On recovering differential systems on a finite interval from spectra. Proc. Steklov Inst. Math. 255, 260–274 (2006). https://doi.org/10.1134/S0081543806040213

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543806040213

Keywords

Navigation