Abstract
Under consideration is a Sturm–Liouville equation with a piecewise entire potential and discontinuity conditions independent of the spectral parameter for the solutions on an unspecified rectifiable curve lying in the complex plane. We study an inverse spectral problem with respect to the ratio of elements of one column or one row of the transfer matrix and give the conditions of uniqueness of a solution. These results are applied to the inverse problem for the Sturm–Liouville equation with piecewise constant complex weight, piecewise entire potential, and discontinuity conditions on a segment.
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References
Levitan B.M., Inverse Sturm–Liouville Problems, VNU Science, Utrecht (1987).
Marchenko V.A., Sturm–Liouville Operators and Applications, Chelsea, Providence (2011).
Poeschel J. and Trubowitz E., Inverse Spectral Theory, Academic, New York (1987).
Yurko V.A., “Boundary value problems with discontinuity conditions in an interior point,” Differ. Equ., vol. 36, no. 8, 1266–1269 (2000).
Yurko V.A., Method of Spectral Mappings in the Inverse Problem Theory, VSP, Utrecht (2002) (Inverse Ill-Posed Probl. Ser.).
Savchuk A.M. and Shkalikov A.A., “Inverse problems for Sturm–Liouville operators with potentials in Sobolev spaces: Uniform stability,” Funct. Anal. Appl., vol. 44, no. 4, 270–285 (2010).
Wasow W., Asymptotic Expansions for Ordinary Differential Equations, Dover, New York (1987).
Fedoryuk M.V., Asymptotic Analysis: Linear Ordinary Differential Equations, Springer, Berlin and Heidelberg (1993).
Ishkin Kh.K., “On localization of the spectrum of the problem with complex weight,” J. Math. Sci. (N. Y.), vol. 150, no. 6, 2488–2499 (2008).
Ishkin Kh.K., “Localization criterion for the spectrum of the Sturm–Liouville operator on a curve,” St. Petersburg Math. J., vol. 28, no. 1, 37–63 (2017).
Ishkin Kh.K. and Rezbayev A.V., “On the Davies formula for the distribution of eigenvalues of a non-self-adjoint differential operator,” J. Math. Sci. (N. Y.), vol. 252, no. 3, 374–383 (2021).
Golubkov A.A., “Asymptotics of transfer matrix of Sturm–Liouville equation with piecewise-entire potential function on a curve,” Moscow Univ. Math. Bull., vol. 74, no. 2, 65–69 (2019).
Golubkov A.A., “A boundary value problem for the Sturm–Liouville equation with piecewise entire potential on the curve and solution discontinuity conditions,” Sib. Electr. Math. Reports, vol. 16, 1005–1027 (2019).
Golubkov A.A., “Spectrum of the Sturm–Liouville operator on a curve with parameters in the boundary conditions and discontinuity conditions for solutions,” in: Modern Methods of the Theory of Boundary Value Problems. Part 4. Proceedings of the Voronezh Spring Mathematical School: “Pontryagin Readings-XXX” (May 3–9, 2019), Tsentr.-Chernozem. Knizh. Izdat., Voronezh (2021), 45–68 [Russian] (Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.; vol. 193).
Ishkin Kh.K., “On a trivial monodromy criterion for the Sturm–Liouville equation,” Math. Notes, vol. 94, no. 4, 508–523 (2013).
Golubkov A.A., “Inverse problem for Sturm–Liouville operators in the complex plane,” Izv. Saratov Univ. Math. Mech. Inform., vol. 18, no. 2, 144–156 (2018).
Golubkov A.A. and Kuryshova Y.V., “Inverse problem for Sturm–Liouville operators on a curve,” Tamkang J. Math., vol. 50, no. 3, 349–359 (2019).
Golubkov A.A., “Inverse problem for the Sturm–Liouville Equation with piecewise entire potential and piecewise constant weight on a curve,” Sib. Electr. Math. Reports, vol. 18, no. 2, 951–974 (2021).
Golubkov A.A. and Makarov V.A., “Reconstruction of dielectric permittivity profile of a plate with strong frequency dispersion,” Moscow Univ. Phys. Bull., vol. 64, no. 6, 658–660 (2009).
Golubkov A.A. and Makarov V.A., “Determining the coordinate dependence of some components of the cubic susceptibility tensor \( \widehat{\chi}^{(3)}(z,\omega,-\omega,\omega,\omega) \) of a one-dimensionally inhomogeneous absorbing plate at an arbitrary frequency dispersion,” Quantum Electron., vol. 40, no. 11, 1045–1050 (2010).
Levitan B.M. and Sargsyan I.S., Sturm–Liouville and Dirac Operators, Kluwer, Dordrecht etc. (1990).
Yurko V.A., Equations of Mathematical Physics, Saratov University, Saratov (2010) [Russian].
Freiling G. and Yurko V.A., “Inverse problems for differential equations with turning points,” Inverse Probl., vol. 13, no. 5, 1247–1263 (1997).
Yurko V.A., “Inverse spectral problems for Sturm–Liouville operators with complex weights,” Inverse Probl. Sci. Engineering, vol. 26, no. 10, 1396–1403 (2018).
Yurko V.A., “On the inverse problem for differential operators on a finite interval with complex weights,” Math. Notes, vol. 105, no. 2, 301–306 (2019).
Duistermaat J.J. and Grünbaum F.A., “Differential equations in the spectral parameter,” Comm. Math. Phys., vol. 103, no. 2, 177–240 (1986).
Coddington E.A. and Levinson N., Theory of Ordinary Differential Equations, McGill-Hill Book, New York, Toronto, and London (1955).
Fikhtengolts G.M., A Course of Differential and Integral Calculus. Vol. 1, Nauka, Moscow (1966) [Russian].
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 3, pp. 485–499. https://doi.org/10.33048/smzh.2023.64.304
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Golubkov, A.A. An Inverse Problem for Sturm–Liouville Operators with a Piecewise Entire Potential and Discontinuity Conditions of Solutions on a Curve. Sib Math J 64, 542–553 (2023). https://doi.org/10.1134/S0037446623030047
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DOI: https://doi.org/10.1134/S0037446623030047