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.
Similar content being viewed by others
References
L.Y. Ainola, An inverse problem on eigenoscillations of elastic covers, Appl. Math. Mech. (1971), 2, 359–365.
F.V Atkinson, Discrete and continuous boundary problems, Academic Press, New York, London, 1964.
R. Beals, The inverse problem for ordinary differential operators on the line, Amer. J. Math. 107(1985), 281–366.
Y.M.Berezanskii, A eigenfunctions expansion for selfadjoint operators, “Naukova Dumka”, Kiev, 1965.
Y.M. Berezanskii, Integration of nonlinear difference equations by inverse spectral problem method, Dokl. Akad. Nauk SSSR, 281 (1985), 16–19.
O.I. Bogoyavlenskii, Integrable dynamic systems connected with the Kd V equation, Izvest. Akad. Nauk SSSR, Ser. Mat. 51(1987), 1123–1141.
P.J. Caudrey, The inverse problem for the third order equation, Phys. Lett. 79A(1980), 264–268.
P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm, Pure Appl. Math. 35(1982), 567–628.
G.S. Guseinov, The determination of the infinite nonselfadjoint Jacobi matrix from its generalized spectral function, Mat. Zametki 23(1978), 237–248.
I.G. Khachatryan, On certain inverse problems for higher-order differential operators on the half-line, Func. Anal. Appl. 17(1983), 40–52.
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.
J.R. McLaughlin, An inverse eigenvalue problems of order four, SIAM J. Math. Anal. 7(1976), 646–661.
M.A. Naimark, Linear differential operators, 2nd ed., “Nauka”, Moscow, 1969; English transi. of 1st ed., Parts I,II, Ungar, New York, 1967, 1968.
F.J. Niordson, A method for solving inverse eigenvalue problems, Recent Progress in Applied Mechanics. The Folk Odquist Volume, Stokholm, 1967, 373–382.
L.A. Sakhnovch, The inverse problem for differential operators of order n > 2 with analytic coefficients, Mat. Sb. 46(88)(1958), 61–76.
.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.
VA. Yurko, Reconstruction of higher-order differential operators, Differentsial’nye Uravneniya 25(1989) 1540–1550; English transl. in Differential Equations 25(1989), 1082-1091.
VA. Yurko, On a certain problem of elacticity, Appl. Math. Mech. 54(1990), 998–1002.
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.
VA. Yurko, Reconstruction of nonselfadjoint differential operators on the half-line from the Weyl matrix, Mat. Sb. 182(1991), 431–456.
VA. Yurko, Solution of the Boussinesq equation on the half- line by the inverse problem method, Inverse Problems, 7(1991), 727–732.
VE. Zacharov, On stochastization of one-dimensional chains of nonlinear oscillators, Zh. Eksp. Teor. Fiz., 65(1973), 219–225.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yurko, V. An inverse problem for operators of a triangular structure. Results. Math. 30, 346–373 (1996). https://doi.org/10.1007/BF03322200
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322200