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Home » Issues » Archive of Issues » 2020 - 2 issue » pp. 237-248

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Azerbaijan Baku
Year
2020
Volume
30
Issue
2
Pages
237-248
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Section Mathematics
Title Spectral properties of the Sturm-Liouville operator with a spectral parameter quadratically included in the boundary condition
Author(-s) Mammadova L.I.a, Nabiev I.M.bcd
Affiliations Azerbaijan State Oil and Industry Universitya, Baku State Universityb, Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciencesc, Khazar Universityd
Abstract The article considers the Sturm-Liouville operator with a real quadratically integrable potential. Boundary conditions are non-separated. One of these boundary conditions includes the quadratic function of the spectral parameter. Some spectral properties of the operator are studied. It is proves that eigenvalues are real and non-zero and there are no associated functions to the eigenfunctions. An asymptotic formula for the spectrum of the operator is derived, and a representation of the characteristic function as an infinite product is obtained. The results of the paper play an important role in solving inverse problems of spectral analysis for differential operators.
Keywords Sturm-Liouville operator, non-separated boundary conditions, eigenvalues, infinite product
UDC 517.98
MSC 34A30
DOI 10.35634/vm200207
Received 7 November 2019
Language Russian
Citation Mammadova L.I., Nabiev I.M. Spectral properties of the Sturm-Liouville operator with a spectral parameter quadratically included in the boundary condition, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 237-248.
References
  1. Collatz L. Eigenwertaufgaben mit technischen Anwendungen, Leipzig: Geest & Portig, 1963.
    Translated under the title Zadachi na sobstvennye znacheniya (s tekhnicheskimi prilozheniyami), Moscow: Nauka, 1968.
  2. Akhtyamov A.M. Teoriya identifikatsii kraevykh uslovii i ee prilozheniya (Identification theory of boundary value problems and its applications), Moscow: Fizmatlit, 2009.
  3. Möller M., Pivovarchik V. Spectral theory of operator pencils, Hermite-Biehler functions, and their applications, Cham: Birkhäuser, 2015. https://doi.org/10.1007/978-3-319-17070-1
  4. Kerimov N.B., Mamedov Kh.R. On one boundary value problem with a spectral parameter in the boundary conditions, Siberian Mathematical Journal, 1999, vol. 40, no. 2, pp. 281-290. https://doi.org/10.1007/s11202-999-0008-5
  5. Atkin A.E., Atkina G.P. A uniqueness theorem for Sturm-Liouville equations with a spectral parameter rationally contained in the boundary condition, Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika, 2011, vol. 4, issue 3, pp. 158-170 (in Russian). http://mi.mathnet.ru/eng/iigum126
  6. Aliev Z.S., Dun'yamalieva A.A. Defect basis property of a system of root functions of a Sturm-Liouville problem with spectral parameter in the boundary conditions, Differential Equations, 2015, vol. 51, issue 10, pp. 1249-1266. https://doi.org/10.1134/S0012266115100018
  7. Shukurov A.Sh. On the number of non-real eigenvalues of the Sturm-Liouville problem, Eurasian Math. J., 2017, vol. 8, no. 3, pp. 77-84. http://mi.mathnet.ru/eng/emj268
  8. Shkalikov A.A. Basis properties of root functions of differential operators with spectral parameter in the boundary conditions, Differential Equations, 2019, vol. 55, issue 5, pp. 631-643. https://doi.org/10.1134/S0012266119050057
  9. Guliyev N.J. Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, Journal of Mathematical Physics, 2019, vol. 60, issue 6, 063501. https://doi.org/10.1063/1.5048692
  10. Megraliev Ya.T., Velieva B.K. Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 166-182 (in Russian). https://doi.org/10.20537/vm190203
  11. Marchenko V.A. Operatory Shturma-Liuvillya i ikh prilozheniya (Sturm-Liouville operators and their applications), Kiev: Naukova Dumka, 1977.
  12. Plaksina O.A. Inverse problems of spectral analysis for Sturm-Liouville operators with nonseparated boundary conditions, Mathematics of the USSR - Sbornik, 1988, vol. 59, no. 1, pp. 1-23. https://doi.org/10.1070/SM1988v059n01ABEH003121
  13. Guseinov I.M., Nabiev I.M. Solution of a class of inverse boundary-value Sturm-Liouville problems, Sbornik: Mathematics, 1995, vol. 186, no. 5, pp. 661-674. https://doi.org/10.1070/SM1995v186n05ABEH000035
  14. Nabiev I.M. Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators, Mathematical Notes, 2000, vol. 67, issue 3, pp. 309-319. https://doi.org/10.1007/BF02676667
  15. Djakov P., Mityagin B.S. Instability zones of periodic 1-dimentional Schrödinger and Dirac operators, Russian Mathematical Surveys, 2006, vol. 61, no. 4, pp. 663-766. https://doi.org/10.1070/RM2006v061n04ABEH004343
  16. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M. The inverse Sturm-Liouville problem with generalized periodic boundary conditions, Doklady Mathematics, 2008, vol. 78, no. 1, pp. 582-584. https://doi.org/10.1134/S1064562408040303
  17. Yurko V. An inverse spectral problem for non-selfadjoint Sturm-Liouville operators with nonseparated boundary conditions, Tamkang Journal of Mathematics, 2012, vol. 43, no. 2, pp. 289-299. https://doi.org/10.5556/j.tkjm.43.2012.1100
  18. Nabiev I.M. Determination of the diffusion operator on an interval, Colloquium Mathematicum, 2014, vol. 134, no. 2, pp. 165-178. https://doi.org/10.4064/cm134-2-2
  19. Baskakov A.G., Polyakov D.M. The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential, Sbornik: Mathematics, 2017, vol. 208, no. 1, pp. 1-43. https://doi.org/10.1070/SM8637
  20. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M. Inverse problem for an operator pencil with nonseparated boundary conditions, Doklady Mathematics, 2009, vol. 79, no. 2, pp. 169-171. https://doi.org/10.1134/S1064562409020069
  21. Yurko V. Inverse problems for non-selfadjoint quasi-periodic differential pencils, Analysis and Mathematical Physics, 2012, vol. 2, pp. 215-230. https://doi.org/10.1007/s13324-012-0030-9
  22. Nabiev I.M., Shukurov A.Sh. Properties of the spectrum and uniqueness of reconstruction of Sturm-Liouville operator with a spectral parameter in the boundary condition, Proceedings of the Institute of Mathematics and Mechanics, 2014, vol. 40, special issue, pp. 332-341. http://proc.imm.az/volumes/40-s/40-s-29.pdf
  23. Ibadzadeh Ch.G., Mammadova L.I., Nabiev I.M. Inverse problem of spectral analysis for diffusion operator with nonseparated boundary conditions and spectral parameter in boundary condition, Azerbaijan Journal of Mathematics, 2019, vol. 9, no. 1, pp. 171-189. http://azjm.org/volumes/0901/pdf/11.pdf
  24. Levin B.Ya. Tselye funktsii (Entire functions), Moscow: Moscow State University, 1971.
  25. Sidorov Yu.V., Fedoryuk M.V., Shabunin M.I. Lektsii po teorii funktsii kompleksnogo peremennogo (Lectures on the theory of functions of a complex variable), Moscow: Nauka, 1982.
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