Inverse problem for Sturm-Liouville operators on a curve
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Abstract
he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.
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How to Cite
Golubkov, A. A., & Kuryshova , Y. V. (2019). Inverse problem for Sturm-Liouville operators on a curve. Tamkang Journal of Mathematics, 50(3), 349–359. https://doi.org/10.5556/j.tkjm.50.2019.3368
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References
1. V. A. Marchenko, Sturm–Liouville Operators and Their Applications, Birkh¨auser, 1986.
2. B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Sci. Press, Utrecht, 1987.
3. V. A. Yurko, Boundary value problems with discontinuity conditions in an interior point of
the interval, Differential Equations, 36:8 (2000), 1266–1269.
4. V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Illposed
Problems Series, VSP, Utrecht, 2002.
5. J. Poeschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, New York, 1987.
6. G. Freiling and V.A. Yurko, Inverse Sturm-Liouville Problems and their Applications. NOVA
Scince Publishers, New York, 2001.
7. A. A. Golubkov and V. A. Makarov, Reconstruction of the coordinate dependence of the
diagonal form of the dielectric permittivity tensor of a one-dimensionally inhomogeneous
medium, Moscow University Physics Bulletin, 65:3 (2010), 189194.
8. A. A. Angeluts, A. A. Golubkov, V. A. Makarov, A. P. Shkurinov, Reconstruction of the
spectrum of the relative permittivity of the plane–parallel plate from the angular dependences
of its transmission coefficients, JETP Letters, 93:4 (2011), 191194.
9. A. A. Golubkov and V. A. Makarov, Inverse spectral problem for a generalized Sturm–Liouville
equation with complex-valued coefficients, Differential Equations, 47:10 (2011), 15141519.
10. B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators, Mathematics and Its
Applications (Soviet Series), v. 59, Kluwer Academic Publishers, 1991.
11. Kh. K. Ishkin, Necessary conditions for the localization of the spectrum of the Sturm–Liouville
problem on a curve, Mathamatical Notes, 78:1 (2005), 6475.
12. R. E. Langer, The boundary problem of an ordinary linear differential system in the complex
domain, Trans. Amer. Math. Soc, 46 (1939), 151190.
13. Kh. K. Ishkin, Localization criterion for the spectrum of the Sturm–Liouville operator on a
curve, St. Petersburg Mathematical Journal, 28:1 (2017), 3763.
14. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, New York, Dover
Publications, 1988.
15. M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-
Verlag, Berlin, 1993.
16. Kh. K. Ishkin, On a trivial monodromy criterion for the Sturm–Liouville equation, Mathamatical
Notes, 94:4 (2013), 508523.
17. A. A. Golubkov, Inverse problem for Sturm–Liouville operators in the complex plane, Izv.
Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:2 (2018), 144156 (in Russian).
18. A. A. Golubkov, Asymptotics of transfer matrix of Sturm–Liouville equation with piecewiseentire
potential function on a curve, Moscow University Mathematics Bulletin, 74:2 (2019)
(Translated from Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 74:2 (2019),
3741).
19. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, New York,
McGraw-Hill, 1955.
20. B. Ja. Levin, Distribution îf Zeros of Entire Functions, Translations of Mathematical Monographs,
v. 5, Amer. Math. Soc., Providence, R.I., 1980
2. B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Sci. Press, Utrecht, 1987.
3. V. A. Yurko, Boundary value problems with discontinuity conditions in an interior point of
the interval, Differential Equations, 36:8 (2000), 1266–1269.
4. V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Illposed
Problems Series, VSP, Utrecht, 2002.
5. J. Poeschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, New York, 1987.
6. G. Freiling and V.A. Yurko, Inverse Sturm-Liouville Problems and their Applications. NOVA
Scince Publishers, New York, 2001.
7. A. A. Golubkov and V. A. Makarov, Reconstruction of the coordinate dependence of the
diagonal form of the dielectric permittivity tensor of a one-dimensionally inhomogeneous
medium, Moscow University Physics Bulletin, 65:3 (2010), 189194.
8. A. A. Angeluts, A. A. Golubkov, V. A. Makarov, A. P. Shkurinov, Reconstruction of the
spectrum of the relative permittivity of the plane–parallel plate from the angular dependences
of its transmission coefficients, JETP Letters, 93:4 (2011), 191194.
9. A. A. Golubkov and V. A. Makarov, Inverse spectral problem for a generalized Sturm–Liouville
equation with complex-valued coefficients, Differential Equations, 47:10 (2011), 15141519.
10. B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators, Mathematics and Its
Applications (Soviet Series), v. 59, Kluwer Academic Publishers, 1991.
11. Kh. K. Ishkin, Necessary conditions for the localization of the spectrum of the Sturm–Liouville
problem on a curve, Mathamatical Notes, 78:1 (2005), 6475.
12. R. E. Langer, The boundary problem of an ordinary linear differential system in the complex
domain, Trans. Amer. Math. Soc, 46 (1939), 151190.
13. Kh. K. Ishkin, Localization criterion for the spectrum of the Sturm–Liouville operator on a
curve, St. Petersburg Mathematical Journal, 28:1 (2017), 3763.
14. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, New York, Dover
Publications, 1988.
15. M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-
Verlag, Berlin, 1993.
16. Kh. K. Ishkin, On a trivial monodromy criterion for the Sturm–Liouville equation, Mathamatical
Notes, 94:4 (2013), 508523.
17. A. A. Golubkov, Inverse problem for Sturm–Liouville operators in the complex plane, Izv.
Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:2 (2018), 144156 (in Russian).
18. A. A. Golubkov, Asymptotics of transfer matrix of Sturm–Liouville equation with piecewiseentire
potential function on a curve, Moscow University Mathematics Bulletin, 74:2 (2019)
(Translated from Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 74:2 (2019),
3741).
19. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, New York,
McGraw-Hill, 1955.
20. B. Ja. Levin, Distribution îf Zeros of Entire Functions, Translations of Mathematical Monographs,
v. 5, Amer. Math. Soc., Providence, R.I., 1980