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Равномерная устойчивость восстановления оператора Штурма–Лиувилля на графе-звезде
1. Kuchment P. Graph models of wave propagation in thin structures // Waves in Random Media. 2002. Vol. 12, no. 4. P. R1–R24. DOI: http://dx.doi.org/10.1088/0959-7174/ 12/4/201
2. Schapotschnikow P., Gnutzmann S. Spectra of graphs and semi-conducting polymers // Analysis on Graphs and Its Applications (Proceedings of Symposia in Pure Mathematics, Vol. 77) / Providence: AMS, 2008. P. 691–705.
3. Баданин А. В., Коротяев Е. Л. Об одном магнитном операторе Шрёдингера на пе-риодическом графе // Математический сборник. 2010. Т. 201, №. 10. С. 3–46. DOI: https://doi.org/10.4213/sm7490.
4. Покорный Ю. В., Пенкин О. М., Прядиев В. Л. и др. Дифференциальные уравнения на геометрических графах. М. : Физматлит, 2004. 268 с.
5. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. Providence: AMS, 2013.270 p. (Mathematical Surveys and Monographs, Vol. 186) DOI: http://dx.doi.org/10.1090/ surv/186
6. Kurasov P. Spectral Geometry of Graphs. Berlin: Springer Nature, 2024. 639 p. (Operator Theory: Advances and Applications, Vol. 293). DOI: https://doi.org/10. 1007/978-3-662-67872-5
7. Yurko V. Inverse spectral problems for Sturm–Liouville operators on graphs. Inverse Problems. 2005. Vol. 21, no. 3. P. 1075. DOI: https://doi.org/10.1088/0266-5611/ 21/3/017
8. Belishev M. I. Boundary spectral inverse problem on a class of graphs (trees) by the BC-method //Inverse Problems. 2004. Vol. 20, no. 3. P. 647–672. DOI: https://doi.org/10. 1088/0266-5611/20/3/002
9. Brown B. M, Weikard R. A Borg-Levinson Theorem for Trees // Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2005. Vol. 461, no. 2062. P. 3231–3243. DOI: 10.1098/rspa.2005.1513. URL: http://www.jstor.org/ stable/30046979 (accessed 27.03.2026)
10. Юрко В. А. Обратные спектральные задачи для дифференциальных операторов на пространственных сетях // Успехи математических наук. 2016. Т. 71, вып. 3(429). С. 149–196. DOI: https://doi.org/10.4213/rm9709
11. Avdonin S. A., Khmelnytskaya K. V., Kravchenko V. V. Reconstruction techniques for quantum trees // Mathematical Methods in the Applied Sciences. 2024. Vol. 47, no. 9. P. 7182–7197. DOI: https://doi.org/10.1002/mma.9963
12. Yurko V. Inverse problems for differential pencils on A-graphs // Journal of Inverse and Ill-posed Problems. 2017. Vol. 25, no. 6. P. 819–828. DOI: https://doi.org/10.1515/ jiip-2016-0065
13. Yurko V. A. Inverse problems for Sturm-Liouville operators on graphs with a cycle // Operators and Matrices. 2008. Vol. 2, no. 4. P. 543–553. DOI: http://dx.doi.org/10. 7153/oam-02-34
14. Freiling G., Ignatiev M. Y., Yurko V. A. An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graph // Analysis on Graphs and Its Applications (Proceedings of Symposia in Pure Mathematics, Vol. 77) / Providence: AMS, 2008. P. 397–408.
15. Ignatyev M. Inverse scattering problem for Sturm–Liouville operators with Bessel singularities on noncompact star-type graphs // Inverse Problems. 2015. Vol. 31, no. 12. P. 125006. DOI: https://doi.org/10.1088/0266-5611/31/12/125006
16. Ignatyev M. Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result // Tamkang Journal of Mathematics. 2015. Vol. 46, no. 4. P. 401–422. DOI: https://doi.org/10.5556/j.tkjm.46.2015.1806
17. Liu D.-Q., Yang C.-F. Inverse spectral problems for Dirac operators on a star graph with mixed boundary conditions // Mathematical Methods in the Applied Sciences. 2021. Vol. 44, iss. 13. P. 10663–10672. DOI: https://doi.org/10.1002/mma.7436
18. Bondarenko N. A partial inverse Sturm-Liouville problem on an arbitrary graph // Mathematical Methods in the Applied Sciences. 2021. Vol. 44, iss. 8. P. 6896–6910. DOI: https://doi.org/10.1002/mma.7231
19. Bondarenko N. P. Partial inverse Sturm-Liouville problems // Mathematics. 2023. Vol. 11, no. 10, article 2408. DOI: https://doi.org/10.3390/math11102408
20. Visco-Comandini F., Mirrahimi M., Sorine M. Some inverse scattering problems on star-shaped graphs // Journal of Mathematical Analysis and Applications. 2011. Vol. 378, iss. 1. P. 343–358. DOI: https://doi.org/10.1016/j.jmaa.2010.12.047
21. Olivieri M., Finco D. On the inverse spectral problems for quantum graphs// Advances in Quantum Mechanics (Springer INdAM Series, Vol. 18) / Cham: Springer, 2017. P. 267–281. DOI: https://doi.org/10.1007/978-3-319-58904-6_16
22. Pivovarchik V. Recovering the shape of an equilateral quantum tree by two spectra // Integral Equations and Operator Theory. 2024. Vol. 96, article 11. DOI: https://doi. org/10.1007/s00020-024-02759-6
23. Kurasov P., Farooq O., L- awniczak M. et al. Families of isospectral and isoscattering quantum graphs // Physical Review Research. 2025. Vol. 7, iss. 2. P. L022071. DOI: https://doi.org/10.1103/6yk9-17y3
24. Buterin S. Functional-differential operators on geometrical graphs with global delay and inverse spectral problems // Results in Mathematics. 2023. Vol. 78, article 79. DOI: https://doi.org/10.1007/s00025-023-01850-5
25. Wang F., Yang C.-F., Buterin S. et al. Inverse spectral problems for Dirac-type operators with global delay on a star graph // Analysis and Mathematical Physics. 2024. Vol. 14, article 24. DOI: https://doi.org/10.1007/s13324-024-00884-4
26. Mochizuki K., Trooshin I. On conditional stability of inverse scattering problem on a lasso-shaped graph //Analysis, Probability, Applications, and Computation (Trends in Mathematics) / Cham: Birkh¨auser, 2019. P. 199–205. DOI: https://doi.org/10.1007/ 978-3-030-04459-6_19
27. Bondarenko N. Stability of the inverse Sturm–Liouville problem on a quantum tree // Studies in Applied Mathematics. 2025. Vol. 155, iss. 6. P. e70162. DOI: https://doi.org/ 10.1111/sapm.70162
28. Bondarenko N. P. Stability of the inverse Sturm–Liouville problem on a graph with a cycle // Journal of Inverse and Ill-posed Problems. 2025. DOI: https://doi.org/10.1515/ jiip-2025-0059
29. Chitorkin E. E., Bondarenko N. P. Uniform stability of the inverse Sturm–Liouville problem on a star–shaped graph // Bolet´in de la Sociedad Matema´tica Mexicana. 2026. Vol. 32, article 53. DOI: https://doi.org/10.1007/s40590-026-00884-3
30. Bondarenko N. P. Spectral data characterization for the Sturm–Liouville operator on the star-shaped graph // Analysis and Mathematical Physics. 2020. Vol. 10, article 83. DOI: https://doi.org/10.1007/s13324-020-00430-y
31. Bondarenko N. P. Constructive solution of the inverse spectral problem for the matrix Sturm–Liouville operator // Inverse Problems in Science and Engineering. 2020. Vol. 28, iss. 9. P. 1307–1330. DOI: https://doi.org/10.1080/17415977.2020.1729760
32. Xu X.-C., Bondarenko N. P. Stability of the inverse scattering problem for the self-adjoint matrix Schro¨dinger operator on the half line // Studies in Applied Mathematics. 2022. Vol. 149, iss. 3. P. 815–838. DOI: https://doi.org/10.1111/sapm.12522
33. Юрко В. А. Обратные спектральные задачи и их приложения. Саратов: Издательство Саратовского педагогического института, 2001. 499 с.
34. Савчук А. М., Шкаликов А. А. Обратные задачи для оператора Штурма–Лиувилля с потенциалами из пространств Соболева. Равномерная устойчивость // Функциональный анализ и его приложения. 2010. Т. 44, вып. 4. С. 34–53. DOI: https://doi.org/10.4213/faa3022
35. Савчук А. М., Шкаликов А.А. Равномерная устойчивость обратной задачи Штурма–Лиувилля по спектральной функции в шкале соболевских пространств // Труды Математического института имени В. А. Стеклова. 2013. Т. 283. С. 188–203. DOI: 10.1134/S0371968513040134. URL: https://www.mathnet.ru/rus/tm3515 (дата доступа 16.04.2026)
36. Buterin S., Kuznetsova M. On Borg’s method for non-selfadjoint Sturm–Liouville operators //Analysis and Mathematical Physics. 2019. Vol. 9, iss. 4. P. 2133–2150. DOI: https://doi.org/10.1007/s13324-019-00307-9
37. Марченко В. А. Операторы Штурма-Лиувилля и их приложения. Киев : Наукова думка, 1977. 331 с.
38. Макаров Б.М., Подкорытов А. Н. Лекции по вещественному анализу. Санкт-Петербург: БХВ-Петербург, 2011. 688 с.
39. Rudin W. Real and Complex Analysis [3-rd edition]. Singapore: McGraw-Hill, 1987. 416 p.
40. Ситник С. М. Операторы преобразования и их приложения // Исследования по современному анализу и математическому моделированию / Владикавказ: Владикавказ. науч. центр РАН и РСО-А, 2008. C. 226–293. URL: https://arxiv.org/pdf/1012.3741 (дата доступа 27.03.2026)
41. Hryniv R. O., Mykytyuk Y. V. Inverse spectral problems for Sturm–Liouville operators with singular potentials // Inverse Problems. 2003. Vol. 19, no. 3. P. 665. DOI: https://doi.org/10.1088/0266-5611/19/3/312
42. Hryniv R. O., Mykytyuk Y. V. Transformation operators for Sturm–Liouville operators with singular potentials // Mathematical Physics, Analysis and Geometry. 2004. Vol. 7. P. 119–149. DOI: https://doi.org/10.1023/B:MPAG.0000024658.58535.74
43. Buterin S. Uniform full stability of recovering convolutional perturbation of the Sturm–Liouville operator from the spectrum // Journal of Differential Equations. 2021. Vol. 282. P. 67–103. DOI: https://doi.org/10.1016/j.jde.2021.02.022
44. Kuznetsova M., Bondarenko N. Addendum to: Solving an inverse problem for the Sturm-Liouville operator with singular potential by Yurko’s method (Tamkang J. Math. 52 (2021), no. 1, 125-154) // Tamkang Journal of Mathematics. 2025. Vol. 56, no. 2. P. 137–139. DOI: https://doi.org/10.5556/j.tkjm.56.2025.5340