Geodesic
On recovering integro-differential operators from the Weyl function
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 276-284 
M. Yu. Ignatiev; S. Yu. Sovetnikova. On recovering integro-differential operators from the Weyl function. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 276-284. http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a3/
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     title = {On recovering integro-differential operators from the {Weyl} function},
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     url = {http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 pp. | MR

[2] Levitan B. M., Inverse Sturm–Liouville problems, VNU Sci. Press, Utrecht, 1987, 246 pp. | MR | Zbl

[3] Freiling G., Yurko V. A., Inverse Sturm–Liouville Problems and their Applications, NOVA Sci. Publ., N. Y., 2001, 305 pp. | MR | Zbl

[4] Beals R., Deift P., Tomei C., Direct and Inverse Scattering on the Line, Math. Surveys and Monographs, 28, Amer. Math. Soc., Providence, RI, 1988, 209 pp. | DOI | MR | Zbl

[5] Yurko V. A., Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002, 316 pp. | MR | Zbl

[6] Yurko V. A., Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000, 265 pp. | MR | Zbl

[7] Lakshmikantham V., Rama Mohana Rao M., Theory of Integro-differential Equations, Stability and Control : Theory and Applications, 1, Gordon and Breach, Singapure, 1995, 308 pp. | MR | Zbl

[8] Yurko V. A., “An inverse problem for integro-differential operators”, Math. Notes, 50:5–6 (1991), 1188–1197 | DOI | MR | Zbl

[9] Kuryshova Yu., “An inverse spectral problem for differential operators with integral delay”, Tamkang J. Math., 42:3 (2011), 295–303 | DOI | MR | Zbl

[10] Buterin S., “On an inverse spectral problem for a convolution integro-differential operator”, Results in Math., 50 (2007), 173–181 | DOI | MR

[11] Buterin S. A., “On the reconstruction of a convolution perturbation of the Sturm–Liouville operator from the spectrum”, Differential Equations, 46:1 (2010), 150–154 | DOI | MR | Zbl