Differential Equations

, Volume 55, Issue 4, pp 500–509 | Cite as

Degenerate Boundary Conditions for the Sturm-Liouville Problem on a Geometric Graph

  • V. A. SadovnichiiEmail author
  • Ya. T. SultanaevEmail author
  • A. M. AkhtyamovEmail author
Ordinary Differential Equations


We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.


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  1. 1.
    Marchenko, V.A., Operatory Shturma-Liuvillya i ikh prilozheniya (Sturm-Liouville Operators and Their Applications), Kiev: Naukova Dumka, 1977.zbMATHGoogle Scholar
  2. 2.
    Shiryaev, E.A, and Shkalikov, A.A., Regular and completely regular differential operators, Math. Notes, 2007, vol. 81, no. 4, pp. 566–570.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Moiseev, E.I. and Kapustin, N.Yu., On the singularities of the root space of a spectral problem with a spectral parameter in the boundary condition, Dokl. Math., 2002, vol. 66, no. 1, pp. 14–18.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Sadovnichii, V.A., Sultanaev, Ya.T., and Akhtyamov, A.M., General inverse Sturm-Liouville problem with symmetric potential, Azerb. J. Math., 2015, vol. 5, no. 2, pp. 96–108.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Stone, M.H., Irregular differential systems of order two and the related expansion problems, Trans. Amer. Math. Soc., 1927, vol. 29, no. 1, pp. 23–53.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sadovnichii, V.A. and Kanguzhin, B.E., A connection between the spectrum of a differential operator with symmetric coefficients and the boundary conditions, Sov. Math. Dokl., 1982, vol. 26, pp. 614–618.zbMATHGoogle Scholar
  7. 7.
    Locker, J., Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators, Fort Collins: Colorado State Univ., 2006.zbMATHGoogle Scholar
  8. 8.
    Makin, A., Two-point boundary-value problems with nonclassical asymptotics on the spectrum, Electron. J. Differ. Equ., 2018, no. 95, pp. 1–7.Google Scholar
  9. 9.
    Akhtyamov, A.M., On the spectrum of an odd-order differential operator, Math. Notes, 2017, vol. 101, no. 5, pp. 755–758.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dezin, A.A., Spectral characteristics of general boundary-value problems for operator D 2, Math. Notes, 1985, vol. 37, no. 2, pp. 142–146.CrossRefzbMATHGoogle Scholar
  11. 11.
    Lang, P. and Locker, J., Spectral theory of two-point differential operators determined by D2. I. Spectral properties, J. Math. Anal. Appl., 1989, vol. 141, pp. 538–558.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Biyarov, B.N. and Dzhumabaev, S.A., A criterion for the Volterra property of boundary value problems for Sturm-Liouville equations, Math. Notes, 1994, vol. 56, no. 1, pp. 751–753.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dzhumabaev, S.A. and Kanguzhin, B.E., On an irregular problem on a finite segment, Izv. Akad. Nauk KazSSR Ser. Fiz.-Mat. Nauk, 1988, no. 1, pp. 14–18.Google Scholar
  14. 14.
    Makin, A.S., On an inverse problem for the Sturm-Liouville operator with degenerate boundary conditions, Differ. Equations, 2014, vol. 50, no. 10, pp. 1402–1406.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Malamud, M.M., On the completeness of the system of root vectors of the Sturm-Liouville operator with general boundary conditions, Funct. Anal. Appl., 2008, vol. 42, no. 3, pp. 45–52.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Akhtyamov, A.M., On degenerate boundary conditions in the Sturm-Liouville problem, Differ. Equations, 2016, vol. 52, no. 8, pp. 1085–1087.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liouville, J., Note sur la théorie de la variation des constantes arbitraires, J. Math. Pures Appl., 1838, vol. 3, no. 1, pp. 342–349.Google Scholar
  18. 18.
    Sansone, G., Equazioni differenziali nel campo reale, Bologna: Zanichelli, 1948, Pt. 1; 1949, Pt. 2. Translated under the title Obyknovennye differentsial’nye uravneniya, Moscow: Inostrannaya Literatura, 1953, Vol. 1; 1954, Vol. 2.zbMATHGoogle Scholar
  19. 19.
    Naimark, M.A., Lineinye differentsial’nye operatory (Linear Differential Operators), Moscow: Nauka, 1969.Google Scholar
  20. 20.
    Akhtyamov, A., Amram, M., and Mouftakhov, A., On reconstruction of a matrix by its minors, Int. J. Math. Educ. Sci. Technol., 2018, vol. 49, no. 2, pp. 268–321.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Mavlyutov Institute of MechanicsUfa Scientific Center of the Russian Academy of SciencesUfaRussia
  3. 3.Akmulla Bashkir State Pedagogical UniversityUfaRussia
  4. 4.Bashkir State UniversityUfaRussia

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