Abstract
We study Sturm–Liouville differential operators on the time scales consisting of finitely many isolated points and closed intervals. In the author’s previous paper, it was established that such operators are uniquely determined by the spectral characteristics of all classical types. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm–Liouville boundary-value problems on time scales with one common boundary condition alternate.
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References
V. A. Marchenko, Sturm–Liouville Operators and Their Applications (Birkhäuser, Basel, 1986).
B. M. Levitan, Inverse Sturm–Liouville Problems (VNU Sci. Press, Utrecht, 1987).
G. Freiling and V. A. Yurko, Inverse Sturm–Liouville Problems and Their Applications (NOVA Science Publishers, New York, 2001).
S. Hilger, “Analysis on measure chains – a unified approach to continuous and discrete calculus,” Results Math. 18 (1-2), 18–56 (1990).
M. Bohner and A. Peterson, Dynamic Equations on Time Scales (Birkhäuser Boston, Boston, MA, 2001).
F. M. Atici, D. C. Biles and A. Lebedinsky, “An application of time scales to economics,” Math. Comput. Modelling 43 (7-8), 718–726 (2006).
K. R. Prasad and Md. Khuddush, “Stability of positive almost periodic solutions for a fishing model with multiple time varying variable delays on time scales,” Bull. Int. Math. Virtual Inst. 9 (3), 521–533 (2019).
S. Ozkan, “Ambarzumyan-type theorems on a time scale,” J. Inverse Ill-Posed Probl. 26 (5), 633–637 (2018).
V. A. Ambarzumyan, “Über eine Frage der Eigenwerttheorie,” Z. Phys. 53, 690–695 (1929).
A. S. Ozkan and I. Adalar, “Half-inverse Sturm–Liouville problem on a time scale,” Inverse Problems 36 (2), 025015 (2020).
S. A. Buterin, M. A. Kuznetsova, and V. A. Yurko, On Inverse Spectral Problem for Sturm–Liouville Differential Operators on Closed Sets, arXiv: 1909.13357 (2019).
V. Yurko, “Inverse problems for Sturm–Liouville differential operators on closed sets,” Tamkang J. Math. 50 (3), 199–206 (2019).
M. Kuznetsova, “A uniqueness theorem on inverse spectral problems for the Sturm–Liouville differential operators on time scales,” Results Math. 75 (2), Paper No. 44 (2020).
F. Atkinson, Discrete and Continuous Boundary Problems (Academic Press, New York, 1964).
F. R. Gantmakher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (American Mathematical Society, Providence, 2002).
G. Sh. Guseinov and H. Tuncay, “On the inverse scattering problem for a discrete one-dimensional Schrödinger equation,” Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist. 44 (1–2), 95–102 (1995).
Ag. Kh. Khanmamedov, “The inverse scattering problem for a perturbed difference Hill equation,” Math. Notes 85 (3), 441–452 (2009).
V. A. Yurko, “An inverse problem for operators of a triangular structure,” Results Math. 30 (3-4), 346–373 (1996).
M. Bohner and H. Koyunbakan, “Inverse problems for Sturm–Liouville difference equations,” Filomat 30, 1297–1304 (2016).
T. Aktosun and V. G. Papanicolaou, “Inverse problem with transmission eigenvalues for the discrete Schrödinger equation,” J. Math. Phys. 56 (8), 082101 (2015).
V. A. Yurko, “Boundary value problems with discontinuity conditions in an interior point of the interval,” Differ. Equations 36 (8), 1266–1269 (2000).
I. M. Guseinov and F. Z. Dostuev, “Inverse Problems for the Sturm–Liouville operator with discontinuity conditions,” Math. Notes 105 (6), 923–928 (2019).
N. P. Bondarenko, “An inverse problem for the non-self-adjoint matrix Sturm–Liouville operator,” Tamkang J. Math. 50 (1), 71–102 (2018).
M. A. Kuznetsova, “Asymptotic formulas for weight numbers of the Sturm–Liouville boundary problem on a star-shaped graph,” Izv. Sarat. Univ. (N. S.), Ser. Mat. Mekh. Inform. 18 (1), 40–48 (2018).
V. A. Yurko, “On recovering Sturm–Liouville operators on graphs,” Math. Notes 79 (4), 572–582 (2006).
R. P. Agarwal, M. Bohner, and P. J. Y. Wong, “Sturm–Liouville eigenvalue problems on time scales,” Appl. Math. Comput. 99 (2–3), 153–166 (1999).
N. Bondarenko, “Recovery of the matrix quadratic differential pencil from the spectral data,” J. Inverse Ill-Posed Probl. 24 (3), 245–263 (2016).
S. A. Buterin and V. A. Yurko, “Inverse problems for second-order differential pencils with Dirichlet boundary conditions,” J. Inverse Ill-Posed Probl. 20 (5-6), 855–881 (2012).
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Kuznetsova, M.A. On Recovering the Sturm–Liouville Differential Operators on Time Scales. Math Notes 109, 74–88 (2021). https://doi.org/10.1134/S0001434621010090
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DOI: https://doi.org/10.1134/S0001434621010090