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An inverse problem for operators of a triangular structure

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Abstract

An inverse problem for operators of a triangular structure is studied. An algorithm for the solution and necessary and sufficient conditions for the solvability of this problem are obtained, moreover uniqueness is proved. Applications to difference and differential operators are considered.

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References

  1. L.Y. Ainola, An inverse problem on eigenoscillations of elastic covers, Appl. Math. Mech. (1971), 2, 359–365.

    Google Scholar 

  2. F.V Atkinson, Discrete and continuous boundary problems, Academic Press, New York, London, 1964.

    MATH  Google Scholar 

  3. R. Beals, The inverse problem for ordinary differential operators on the line, Amer. J. Math. 107(1985), 281–366.

    Article  MathSciNet  MATH  Google Scholar 

  4. Y.M.Berezanskii, A eigenfunctions expansion for selfadjoint operators, “Naukova Dumka”, Kiev, 1965.

  5. Y.M. Berezanskii, Integration of nonlinear difference equations by inverse spectral problem method, Dokl. Akad. Nauk SSSR, 281 (1985), 16–19.

    MathSciNet  Google Scholar 

  6. O.I. Bogoyavlenskii, Integrable dynamic systems connected with the Kd V equation, Izvest. Akad. Nauk SSSR, Ser. Mat. 51(1987), 1123–1141.

    Google Scholar 

  7. P.J. Caudrey, The inverse problem for the third order equation, Phys. Lett. 79A(1980), 264–268.

    MathSciNet  Google Scholar 

  8. P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm, Pure Appl. Math. 35(1982), 567–628.

    Article  MathSciNet  MATH  Google Scholar 

  9. G.S. Guseinov, The determination of the infinite nonselfadjoint Jacobi matrix from its generalized spectral function, Mat. Zametki 23(1978), 237–248.

    MathSciNet  Google Scholar 

  10. I.G. Khachatryan, On certain inverse problems for higher-order differential operators on the half-line, Func. Anal. Appl. 17(1983), 40–52.

    MathSciNet  MATH  Google Scholar 

  11. Z.L. Leibenzon, An inverse problem of spectral analysis of ordinary differential operators of higher order, Trudy Mosk. Mat. Obshch. 15(1966), 70–144; English transl. Moscow Math. Soc. 15(1966), 78-163.

    MathSciNet  Google Scholar 

  12. J.R. McLaughlin, An inverse eigenvalue problems of order four, SIAM J. Math. Anal. 7(1976), 646–661.

    Article  MathSciNet  MATH  Google Scholar 

  13. M.A. Naimark, Linear differential operators, 2nd ed., “Nauka”, Moscow, 1969; English transi. of 1st ed., Parts I,II, Ungar, New York, 1967, 1968.

    Google Scholar 

  14. F.J. Niordson, A method for solving inverse eigenvalue problems, Recent Progress in Applied Mechanics. The Folk Odquist Volume, Stokholm, 1967, 373–382.

    Google Scholar 

  15. L.A. Sakhnovch, The inverse problem for differential operators of order n > 2 with analytic coefficients, Mat. Sb. 46(88)(1958), 61–76.

    MathSciNet  Google Scholar 

  16. .VA. Yurko, Uniqueness of the reconstruction of twoterm differential operators from two spectra, Mat. Zamerki 43(1988), 356–364 English transl, in Math. Notes 43(1988), 3-4, 205-210.

    MathSciNet  MATH  Google Scholar 

  17. VA. Yurko, Reconstruction of higher-order differential operators, Differentsial’nye Uravneniya 25(1989) 1540–1550; English transl. in Differential Equations 25(1989), 1082-1091.

    MathSciNet  Google Scholar 

  18. VA. Yurko, On a certain problem of elacticity, Appl. Math. Mech. 54(1990), 998–1002.

    MathSciNet  Google Scholar 

  19. VA. Yurko, Recovery of differential operators from the Weyl matrix, Dokl. Akad. Nauk SSSR, 313(1990),1368–1372; English transi, in Soviet Math. Dokl. 42(1991), 229-233.

    Google Scholar 

  20. VA. Yurko, Reconstruction of nonselfadjoint differential operators on the half-line from the Weyl matrix, Mat. Sb. 182(1991), 431–456.

    MATH  Google Scholar 

  21. VA. Yurko, Solution of the Boussinesq equation on the half- line by the inverse problem method, Inverse Problems, 7(1991), 727–732.

    Article  MathSciNet  Google Scholar 

  22. VE. Zacharov, On stochastization of one-dimensional chains of nonlinear oscillators, Zh. Eksp. Teor. Fiz., 65(1973), 219–225.

    Google Scholar 

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Yurko, V. An inverse problem for operators of a triangular structure. Results. Math. 30, 346–373 (1996). https://doi.org/10.1007/BF03322200

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