Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Yurko V. A. On Inverse Periodic Problem for Differential Operators for Central Symmetric Potentials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 1, pp. 68-75. DOI: 10.18500/1816-9791-2016-16-1-68-75, EDN: VUSOER

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On Inverse Periodic Problem for Differential Operators for Central Symmetric Potentials

Yurko Vyacheslav Anatol'evich, Saratov State University

An inverse spectral problem for Sturm–Liouville operators on a finite interval with periodic boundary conditions is studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss the statement of the problem, provide an algorithm for its solution along with necessary and sufficient conditions for the solvability of this nonlinear inverse problem.

  1. Marchenko V. A. Sturm – Liouville operators and their applications. Birkhauser, 1986.
  2. Levitan B. M. Inverse Sturm – Liouville problems. Utrecht, VNU Sci. Press, 1987.
  3. Poschel J., Trubowitz E. ¨ Inverse Spectral Theory. New York, Academic Press, 1987.
  4. Freiling G., Yurko V. A. Inverse Sturm – Liouville Problems and their Applications. New York, NOVA Science Publ., 2001.
  5. Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-posed Problems Series. Utrecht, VSP, 2002.
  6. Stankevich I. V. An inverse problem of spectral analysis for Hill’s equation. Soviet Math. Dokl., 1970, vol. 11, pp. 582–586.
  7. Marchenko V. A., Ostrovskii I. V. A characterization of the spectrum of the Hill operator. Math. USSR-Sb., 1975, vol. 26, no. 4, pp. 493–554. DOI: https://doi.org/10.1070/SM1975v026n04ABEH002493.
  8. Yurko V. A. An inverse problem for second order differential operators with regular boundary conditions. Math. Notes, 1975, vol. 18, no. 3–4, pp. 928– 932. DOI: https://doi.org/10.1007/BF01153046.
  9. Yurko V. A. On a periodic boundary value problem. Differ. Equations and Theory of Functions, Saratov, Saratov Univ. Press, 1981, pp. 109–115 (in Russian).
  10. Yurko V. A. On recovering differential operators with nonseparated boundary conditions. Study in Math. and Appl., Ufa, Bashkir Univ. Press, 1981, pp. 55–58 (in Russian).
  11. Plaksina O. A. Inverse problems of spectral analysis for the Sturm – Liouville operators with nonseparated boundary conditions. Math. USSRSb., 1988, vol. 59, no. 1, pp. 1–23. DOI: https://doi.org/10.1070/SM1988v059n01ABEH003121.
  12. Guseinov I. M., Gasymov M. G., Nabiev I. M. An inverse problem for the Sturm – Liouville operator with nonseparable self-adjoint boundary conditions. Siberian Math. J., 1990, vol. 31, no. 6, pp. 910–918.
  13. Guseinov I. M., Nabiev I. M. Solution of a class of inverse boundary-value Sturm – Liouville problems. Sb. Math., 1995, vol. 186, no. 5, pp. 661–674. DOI: https://doi.org/10.1070/SM1995v186n05ABEH000035.
  14. Kargaev P., Korotyaev E. The inverse problem for the Hill operator, a direct approach. Invent. Math., 1997, vol. 129, no. 3, pp. 567–593.
  15. Yurko V. A. On differential operators with nonseparated boundary conditions. Funct. Anal. Appl., 1994, vol. 28, no. 4, pp. 295–297. DOI: https://doi.org/10.1007/BF01076118.
  16. Yurko V. A. The inverse spectral problem for differential operators with nonseparated boundary conditions. J. Math. Analysis Appl., 2000, vol. 250, no. 1, pp. 266–289.
  17. Freiling G., Yurko V. A. On the stability of constructing a potential in the central symmetry case. Applicable Analysis, 2011, vol. 90, no. 12, pp. 1819–1828.