Differential Equations

, Volume 55, Issue 1, pp 24–33 | Cite as

Inverse Spectral Problems for Differential Pencils on Arbitrary Compact Graphs

  • V. YurkoEmail author
Ordinary Differential Equations


Inverse problems of spectral analysis are studied for second-order differential pencils on arbitrary compact graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pokorny, Yu.V., Penkin, O.M., Pryadiev, V.L., Borovskikh A.V., Lazarev, K.P., Shabrov, C.A., Differentsial’nye uravneniya na geometricheskikh grafakh (Differential Equations on Geometric Graphs), Moscow: Fizmatlit, 2004.Google Scholar
  2. 2.
    Marchenko, V.A., Operatory Shturma–Liuvillya i ikh prilozheniya, Kiev: Naukova Dumka, 1977. Translated under the title Sturm–Liouville Operators and Their Applications, Birkhäuser, 1986.Google Scholar
  3. 3.
    Levitan, B.M., Obratnye zadachi Shturma–Liuvillya, Moscow: Nauka, 1984. Translated under the title Inverse Sturm–Liouville Problems, Utrecht: VNU Sci. Press, 1987.zbMATHGoogle Scholar
  4. 4.
    Freiling, G. and Yurko, V.A., Inverse Sturm–Liouville Problems and Their Applications, New York: NOVA Science Publishers, 2001.zbMATHGoogle Scholar
  5. 5.
    Yurko, V.A., Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-Posed Problems Series, Utrecht: VSP, 2002.CrossRefzbMATHGoogle Scholar
  6. 6.
    Yurko, V.A., Vvedenie v teoriyu obratnykh spektral’nykh zadach (Introduction to the Theory of Inverse Spectral Problems), Moscow: Fizmatlit, 2007.Google Scholar
  7. 7.
    Belishev, M.I., Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 2004, vol. 20, pp. 647–672.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Yurko, V.A., Inverse spectral problems for Sturm–Liouville operators on graphs, Inverse Problems, 2005, vol. 21, no. 3, pp. 1075–1086.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brown, B.M. and Weikard, R., A Borg–Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2005, vol. 461, no. 2062, pp. 3231–3243.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yang, C-Fu and Yang, X-P., Uniqueness theorems from partial information of the potential on a graph, J. Inverse Ill-Posed Problems, 2011, vol. 19, no. 4–5, pp. 631–639.CrossRefzbMATHGoogle Scholar
  11. 11.
    Yurko, V.A., Reconstruction of Sturm–Liouville differential operators on A-graphs, Differ. Equations, 2011, vol. 47, no. 1, pp. 50–59.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yurko, V.A., Inverse spectral problems for differential operators on arbitrary compact graphs, J. Inverse Ill-Posed Proplems, 2010, vol. 18, no. 3, pp. 245–261.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Yurko, V.A., An inverse problem for differential pencils on graphs with a cycle, J. Inverse Ill-Posed Problems, 2014, vol. 22, no. 5, pp. 625–641.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yurko, V.A., Inverse problem for differential pencils on hedgehog-graphs, Differ. Equations, 2016, vol. 52, no. 3, pp. 335–345.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mennicken, R. and Möller, M., Non-Self-Adjoint Boundary Value Problems, North-Holland Mathematic Studies, Vol. 192, Amsterdam: North-Holland, 2003.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSaratov State UniversitySaratovRussia

Personalised recommendations