On Recovering Integro-Differential Operators from the Weyl Function


We study inverse problems of spectral analysis for second order integro-differential operators, which are a perturbation of the Sturm–Liouville operator by the integral Volterra operator. We pay the main attention to the nonlinear inverse problem of recovering the potential from the given Weyl function provided that the kernel of the integral operator is known a priori. We obtain properties of the spectral characteristics and the Weyl function, provide an algorithm for constructing the solution of the inverse problem and establish the uniqueness of the solution. For solving the inverse problem we use the method of standard models.


1. Marchenko V. A. Sturm–Liouville operators and their applications. Birkhäuser, 1986. 393 p. (Russ. ed. : Kiev, Naukova Dumka, 1977. 393 p.)
2. Levitan B. M. Inverse Sturm–Liouville problems. Utrecht, VNU Sci. Press, 1987. 246 p. (Russ. ed. : Moscow, Nauka, 1984. 246 p.)
3. Freiling G., Yurko V. A. Inverse Sturm–Liouville Problems and their Applications. New York, NOVA Sci. Publ., 2001. 305 p.
4. Beals R., Deift P., Tomei C. Direct and Inverse Scattering on the Line. (Math. Surveys and Monographs, vol. 28), Providence, RI, Amer. Math. Soc., 1988. 209 p.
5. Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-posed Problems Series. Utrecht, VSP, 2002. 316 p.
6. Yurko V. A. Inverse Spectral Problems for Differential Operators and their Applications. Amsterdam, Gordon and Breach, 2000. 265 p.

7. Lakshmikantham V., Rama Mohana Rao M. Theory of integro-differential equations. (Stability and Control: Theory and Applications, vol.1), Singapure, Gordon and Breach, 1995. 308 p.
8. Yurko V. A. An inverse problem for integro-differential operators. Math. Notes, 1991, vol. 50, iss. 5–6, pp. 1188–1197.
9. Kuryshova Yu. An inverse spectral problem for differential operators with integral delay. Tamkang J. Math., 2011, vol. 42, no. 3, pp. 295–303.
10. Buterin S. A. On an inverse spectral problem for a convolution integro-differential operator. Results in Math., 2007, vol. 50, pp. 173–181.
11. Buterin S. A. On the reconstruction of a convolution perturbation of the Sturm–Liouville operator from the spectrum. Differential Equations, 2010, vol. 46, no. 1, pp. 150 154.

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