Functional Analysis and Its Applications

, Volume 39, Issue 2, pp 148–151 | Cite as

Ambarzumian’s Theorem for a Sturm-Liouville Boundary Value Problem on a Star-Shaped Graph

  • V. N. Pivovarchik


Ambarzumian’s theorem describes the exceptional case in which the spectrum of a single Sturm-Liouville problem on a finite interval uniquely determines the potential. In this paper, an analog of Ambarzumian’s theorem is proved for the case of a Sturm-Liouville problem on a compact star-shaped graph. This case is also exceptional and corresponds to the Neumann boundary conditions at the pendant vertices and zero potentials on the edges.

Key words

inverse problem Neumann boundary conditions normal eigenvalue multiplicity of an eigenvalue least eigenvalue minimax principle 


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  1. 1.
    V. Ambarzumian, Z. Phys., 53, 690–695 (1929).Google Scholar
  2. 2.
    G. Borg, Acta Math., 78, 1–96 (1946).Google Scholar
  3. 3.
    B. M. Levitan and M. G. Gasymov, Usp. Mat. Nauk, 19(2, 3–63 (1964).Google Scholar
  4. 4.
    V. A. Marchenko, Sturm-Liouville Operators and Applications [in Russian], Naukova Dumka, Kiev, 1977; English transl.: Birkhauser, Basel-Boston, Mass., 22 (1986), 1977.Google Scholar
  5. 5.
    N. V. Kuznetsov, Dokl. Akad. Nauk SSSR, 146, 1259–1262 (1962).Google Scholar
  6. 6.
    E. M. Harrel, Amer. J. Math. 109(5), 787–795 (1987).Google Scholar
  7. 7.
    H.-H. Chern and C.-L. Shen, Inverse Problems, 13, 15–18 (1997).Google Scholar
  8. 8.
    P. Exner, Lett. Math. Phys., 38, 313–320 (1996).Google Scholar
  9. 9.
    Yu. Melnikov and B. Pavlov, J. Math. Phys., 42, No.3, 1202–1228 (2001).Google Scholar
  10. 10.
    N. I. Gerasimenko, Teor. Mat. Fiz., 75, No.2, 187–200 (1988).Google Scholar
  11. 11.
    R. Carlson, Trans. Amer. Math. Soc., 351, No.10, 4069–4088 (1999).Google Scholar
  12. 12.
    V. Pivovarchik, SIAM J. Math. Anal., 32, No.4, 801–819 (2000).Google Scholar
  13. 13.
    J. von Below, In: Partial Differential Equations on Multistructures, Lect. Notes Pure Math., Vol. 219, M. Dekker, NY, 2001, pp. 19–36.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. N. Pivovarchik
    • 1
  1. 1.South-Ukrainian State Pedagogical UniversityOdessaUkraine

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