Geodesic
On recovering differential operators on a closed set from spectra
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 19 (2019) no. 4, pp. 389-396 
V. A. Yurko. On recovering differential operators on a closed set from spectra. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 19 (2019) no. 4, pp. 389-396. http://geodesic.mathdoc.fr/item/ISU_2019_19_4_a2/
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     title = {On recovering differential operators on a closed set from spectra},
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     url = {http://geodesic.mathdoc.fr/item/ISU_2019_19_4_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Sturm–Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points. In order to solve the inverse spectral problem for this class of closed sets, we develop ideas of the method of spectral mappings. We also establish and use connections between the Weyl-type functions related to different subsets of the main closed set. Using these ideas and properties we obtain a global constructive procedure for the solution of the nonlinear inverse problem considered, and we establish the uniqueness of the solution of the inverse problem.

[1] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, MA, 2001, 358 pp. | MR | Zbl

[2] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, MA, 2003, 348 pp. | MR | Zbl

[3] V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, Basel, 1986, 393 pp. | MR | Zbl

[4] B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Science Press, Utrecht, 1987, 246 pp. | MR | Zbl

[5] G. Freiling, V. A. Yurko, Inverse Sturm–Liouville Problems and Their Applications, NOVA Science Publ., New York, 2001, 305 pp. | MR | Zbl

[6] V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-Posed Problems Series, 31, VSP, Utrecht, 2002, 306 pp. | DOI | MR

[7] J. R. McLaughlin, “Analytical methods for recovering coefficients in differential equations from spectral data”, SIAM Rev., 28 (1986), 53–72 | DOI | MR | Zbl

[8] J. Poeschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, 1987, 192 pp. | MR | Zbl

[9] R. Beals, P. Deift, C. Tomei, Direct and inverse scattering on the line, Mathematical Surveys and Monographs, 28, AMS, Providence, RI, 1988, 209 pp. | DOI | MR | Zbl

[10] V. G. Romanov, Inverse Problems in Mathematical Physics, VNU Science Press, Utrecht, 1987, 367 pp. | MR

[11] L. A. Sakhnovich, Spectral Theory of Canonical Differential Systems. Method of Operator Identities, Operator Theory: Advances and Applications, 107, Birkhäuser, Basel, 1999, 408 pp. | DOI | MR | Zbl

[12] V. A. Yurko, “Inverse spectral problems for Sturm–Liouville operators on graphs”, Inverse Problems, 21:3 (2005), 1075–1086 | DOI | MR | Zbl

[13] D. Arov, H. Dym, Bitangential direct and inverse problems for systems of integral and differential equations, Encyclopedia of Mathematics and its Applications, 145, Cambridge University Press, 2012, 472 pp. | DOI | MR | Zbl

[14] V. Pivovarchik, M. Moeller, Spectral Theory of Operator Pencils, Hermite–Biehler Functions and their Applications, Operator Theory: Advances and Applications, 246, Birkhaüser, Cham, 2015, 412 pp. | DOI | MR | Zbl

[15] V. A. Yurko, “Inverse spectral problems for differential operators on spatial networks”, Russian Mathematical Surveys, 71:3 (2016), 539–584 | DOI | MR | Zbl