ON AN ANALOGUE OF AMBARZUMYAN THEOREM

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Abstract

For the Sturm–Liouville operator on the half-axis with a complex decreasing potential that allows analytical continuation to some neighborhood of zero, an analogue of Ambarzumyan’s theorem is obtained.

About the authors

Kh. K Ishkin

Ufa University of Science and Technology

Email: ishkin62@mail.ru
Russia

References

  1. Ambarzumian, V.A. Uberline Frage der Eigenwerttheorie / V.A. Ambarzumian // Zeitschrift fur Physik. — 1929. — Bd. 53. — S. 690–695.
  2. Borg, G. Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe / G. Borg // Acta Math. — 1946. — Bd. 78. — S. 1–96.
  3. Kuznetsov, N.V., A generalization of a theorem stated by V.A. Ambartsumyan, Sov. Math., Dokl., 1963, vol. 3, pp. 1475–1478.
  4. Harrel, E.M. On the extension of Ambarzumian’s inverse spectral theorem to compact symmetric spaces / E.M. Harrel // Amer. J. Math. — 1987. — V. 109, № 5. — P. 787–795.
  5. Chakravarty, N.K. On an extension of the theorem of V.A. Ambarzumyan / N.K. Chakravarty, S.K. Acharyya // Proc. R. Soc. Edinb. — 1988. — V. 110A. — P. 79–84.
  6. Chern, H.-H. On the 𝑛-dimensional Ambarzumyan’s theorem / H.-H. Chern, C.-L. Shen // Inverse Problems. — 1997. — V. 13. — P. 15–18.
  7. Levitan, B.M. and Gasymov, M.G., Determination of a differential equation by two of its spectra, Russ. Math. Surv., 1964, vol. 19, no. 2, pp. 1–63.
  8. Rid, M. and Simon, B., Methods of Modern Mathematical Physics. Vol. 4: Operators Theory, New York; San Francisco; London: Academic Press, 1978.
  9. Horv´ats, M. On a theorem of Ambarzumyan / M. Horv´ats // Proc. R. Soc. Edinb. — 2001. — V. 131A. — P. 899–907.
  10. Ishkin, Kh.K., On the spectral instability of the Sturm–Liouville operator with a complex potential, Differ. Equat., 2009, vol. 45, no. 4, pp. 494–509.
  11. Davies, E.B. Wild spectral behaviour on anharmonic oscillators / E.B. Davies // Bull. London Math. Soc. — 2000. — V. 32, № 4. — P. 432–438.
  12. Lidskii, V.B., A non-self-adjoint operator of Sturm–Liouville type with discrete spectrum, Trans. Mosc. Math. Soc., 1960, vol. 9, pp. 45–79.
  13. Kato, T., Perturbation Theory for Linear Operators, Berlin; Heidelberg; New York: Springer-Verlag, 1966.
  14. Naimark, M.A., Linear Differential Operators. Vol. I, II, New York: Frederick Ungar Publishing Co., 1967, 1968.
  15. Pavlov, B. S., The non-selfadjoint operator −𝑦′′ +𝑞(𝑥)𝑦 on a half-line, Sov. Math., Dokl., 1962, vol. 2, pp. 1565–1568.
  16. Pavlov, B.S., On the spectral theory of non-selfadjoint differential operators, Sov. Math., Dokl., 1962, vol. 3, pp. 1483–1487.
  17. Ishkin, Kh.K., On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential, Ufa Math. J., 2013, vol. 5, no. 1, pp. 36–55.
  18. Привалов, И.И. Граничные свойства аналитических функций / И.И. Привалов. — М.–Л. : ГИТТЛ, 1950. — 336 c.
  19. Ishkin, Kh.K. On continuity of the spectrum of a singular quasi-differential operator with respect to a parameter / Kh.K. Ishkin // Eurasian Math. J. — 2011. — V. 2, № 3. — P. 67–81.
  20. Ishkin, Kh.K., A localization criterion for the spectrum of the Sturm–Liouville operator on a curve, St. Petersburg Math. J., 2017, vol. 28, no. 1, pp. 37–63.

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