Skip to main content
SpringerLink
Log in

An inverse sturm-Liouville problem by three spectra

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

Abstract

It is shown that the potential of the Sturm-Liouville equation on interval [0,a] may be restored by the spectra of three boundary problems generated by the equation on the intervals [0,a], [0, 1/2a] and [1/2a,a], respectively. The algorithm of construction is given as well as the sufficient conditions for three sequences of real numbers to be the spectra of the mentioned boundary problems. The problem on [0,a] describes small vibrations of a smooth string with fixed ends. The problems on the half-intervals describe vibrations of the same string clamped at the point of equilibrium.

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

  • [CH] R.Courant, D.Hilbert. Methods of Mathematical Physics, Vol.1, Interscience, New York, (1953).

    Google Scholar 

  • [GK] F.R.Gantmacher, M.G.Krein. Oscillating matrices and kernels and vibrations of mechanical systems (in Russian), GITTL, Moscow-Leningrad, (1950), German transl. Akademie Verlag, Berlin, (1960).

    Google Scholar 

  • [GS] F.Gesztesy, B.Simon. Inverse Spectral Analysis With Partial Information on the Potential, II. The Case of Discrete Spectrum, to be published.

  • [H] O.Hald. Discontinuous inverse eigenvalue problems. Communs. on Pure and Appl. Math., Vol.37 (1984), P.537–577.

    Google Scholar 

  • [HL] H.Hochstadt, B.Lieberman. An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., Vol.34 (1978), P.676–680.

    Google Scholar 

  • [LL] B.Ja.Levin, Yu.I.Lyubarskii, Interpolation by entire functions of special classes and related expansions in series of exponents (in Russian), Izv. Acad. Sci. USSR, Vol.39, No.3 (1975), P.657–702.

    Google Scholar 

  • [LO] B.Ja.Levin, I.V.Ostrovskii, On small perturbations of sets of roots of sinus-type functions (in Russian), Izv. Akad. Nauk USSR, ser. Mat. Vol.43, (1979), p.87–110.

    Google Scholar 

  • [L] B.M.Levitan, Inverse Sturm-Liouville problems (in Russian), Moscow, Nauka (1984), 240 pp.

    Google Scholar 

  • [LG] B.M.Levitan, M.G.Gasymov, Determination of differential equation by two spectra (in Russian), Uspechi Math. Nauk, Vol.19, N 2/116 (1964), p.3–63.

    Google Scholar 

  • [M] V.A.Marchenko, Sturm-Liouville operators and applications, Birkhauser, OT, 22, 1986, 367 pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Higher Mathematics, Odessa Academy of Civil Engineering and Architecture, Didrihson str. 4, 270029, Odessa, Ukraine

    Vyacheslav N. Pivovarchik

Authors
  1. Vyacheslav N. Pivovarchik
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pivovarchik, V.N. An inverse sturm-Liouville problem by three spectra. Integr equ oper theory 34, 234–243 (1999). https://doi.org/10.1007/BF01236474

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01236474

MSC Primary

Search

Navigation