Incomplete inverse problem for Dirac operator with constant delay

Wang, Feng; Yang, Chuan-Fu

doi:10.1090/proc/16736

Incomplete inverse problem for Dirac operator with constant delay
HTML articles powered by AMS MathViewer

by Feng Wang and Chuan-Fu Yang;

Proc. Amer. Math. Soc. 152 (2024), 1561-1572

DOI: https://doi.org/10.1090/proc/16736

Published electronically: February 14, 2024

HTML | PDF | Request permission

Abstract:

In this work, we consider Dirac-type operators with a constant delay less than two-fifths of the interval and not less than one-third of the interval. For our considered Dirac-type operators, an incomplete inverse spectral problem is studied. Specifically, when two complex potentials are known a priori on a certain subinterval, reconstruction of the two potentials on the entire interval is studied from complete spectra of two boundary value problems with one common boundary condition. The uniqueness of the solution of the inverse problem is proved. A constructive method is developed for the solution of the inverse problem.

References

S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Inverse spectral problems for Dirac operators with summable potentials, Russ. J. Math. Phys. 12 (2005), no. 4, 406–423. MR 2201307
N. Bondarenko and V. Yurko, An inverse problem for Sturm-Liouville differential operators with deviating argument, Appl. Math. Lett. 83 (2018), 140–144. MR 3795682, DOI 10.1016/j.aml.2018.03.025
Natalia P. Bondarenko and Vjacheslav A. Yurko, Partial inverse problems for the Sturm-Liouville equation with deviating argument, Math. Methods Appl. Sci. 41 (2018), no. 17, 8350–8354. MR 3891294, DOI 10.1002/mma.5265
S. A. Buterin, Functional-differential operators on geometrical graphs with global delay and inverse spectral problems, Results Math. 78 (2023), Paper No. 79.
S. Buterin and N. Djurić, Inverse problems for Dirac operators with constant delay: uniqueness, characterization, uniform stability, Lobachevskii J. Math. 43 (2022), no. 6, 1492–1501. MR 4491818, DOI 10.1134/S1995080222090050
S. A. Buterin, M. A. Malyugina, and C.-T. Shieh, An inverse spectral problem for second-order functional-differential pencils with two delays, Appl. Math. Comput. 411 (2021), Paper No. 126475, 19. MR 4285852, DOI 10.1016/j.amc.2021.126475
S. A. Buterin, M. Pikula, and V. A. Yurko, Sturm-Liouville differential operators with deviating argument, Tamkang J. Math. 48 (2017), no. 1, 61–71. MR 3623427, DOI 10.5556/j.tkjm.48.2017.2264
S. A. Buterin and V. A. Yurko, An inverse spectral problem for Sturm-Liouville operators with a large constant delay, Anal. Math. Phys. 9 (2019), no. 1, 17–27. MR 3933524, DOI 10.1007/s13324-017-0176-6
Nebojša Djurić and Sergey Buterin, On an open question in recovering Sturm-Liouville-type operators with delay, Appl. Math. Lett. 113 (2021), Paper No. 106862, 6. MR 4173235, DOI 10.1016/j.aml.2020.106862
Nebojša Djurić and Sergey Buterin, On non-uniqueness of recovering Sturm-Liouville operators with delay, Commun. Nonlinear Sci. Numer. Simul. 102 (2021), Paper No. 105900, 6. MR 4268681, DOI 10.1016/j.cnsns.2021.105900
Nebojša Djurić and Sergey Buterin, Iso-bispectral potentials for Sturm-Liouville-type operators with small delay, Nonlinear Anal. Real World Appl. 63 (2022), Paper No. 103390, 10. MR 4290301, DOI 10.1016/j.nonrwa.2021.103390
Nebojša Djurić and Vladimir Vladičić, Incomplete inverse problem for Sturm-Liouville type differential equation with constant delay, Results Math. 74 (2019), no. 4, Paper No. 161, 13. MR 3998756, DOI 10.1007/s00025-019-1087-7
Nebojša Djurić and Biljana Vojvodić, Inverse problem for Dirac operators with a constant delay less than half the length of the interval, Appl. Anal. Discrete Math. 17 (2023), no. 1, 249–261. MR 4602125, DOI 10.2298/AADM221211009D
G. Freiling and V. A. Yurko, Inverse problems for Sturm-Liouville differential operators with a constant delay, Appl. Math. Lett. 25 (2012), no. 11, 1999–2004. MR 2957794, DOI 10.1016/j.aml.2012.03.026
M. G. Gasymov and T. T. Džabiev, Solution of the inverse problem by two spectra for the Dirac equation on a finite interval, Akad. Nauk Azerbaĭdžan. SSR Dokl. 22 (1966), no. 7, 3–6 (Russian, with Azerbaijani summary). MR 203138
M. G. Gasymov and B. M. Levitan, The inverse problem for the Dirac system, Dokl. Akad. Nauk SSSR 167 (1966), 967–970 (Russian). MR 194650
Oleg Gorbunov and Viacheslav Yurko, Inverse problem for Dirac system with singularities in interior points, Anal. Math. Phys. 6 (2016), no. 1, 1–29. MR 3459304, DOI 10.1007/s13324-015-0097-1
R. O. Hryniv, Analyticity and uniform stability in the inverse spectral problem for Dirac operators, J. Math. Phys. 52 (2011), 063513.
M. Yu. Ignatiev, On an inverse Regge problem for the Sturm-Liouville operator with deviating argument, J. Samara State Tech. Univ. Ser. Phys. Math. Sci. 22 (2018) no. 2, 203–211.
B. M. Levitan and I. S. Sargsyan, Operatory Shturma–Liuvillya i Diraka, “Nauka”, Moscow, 1988 (Russian). MR 958344
Alexander Makin, On the spectrum of non-self-adjoint Dirac operators with quasi-periodic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 153 (2023), no. 4, 1099–1117. MR 4611212, DOI 10.1017/prm.2022.33
M. M. Malamud, Uniqueness questions in inverse problems for systems of differential equations on a finite interval, Trans. Mosc. Math. Soc. 60 (1999), 204–262.
M. Pikula, Determination of a Sturm-Liouville-type differential operator with delay argument from two spectra, Mat. Vestn. 43 (1991), no. 3–4, 159–171.
Vladimir Vladičić and Milenko Pikula, An inverse problems for Sturm-Liouville-type differential equation with a constant delay, Sarajevo J. Math. 12(24) (2016), no. 1, 83–88. MR 3511149, DOI 10.5644/SJM.12.1.06
Milenko Pikula, Vladimir Vladičić, and Biljana Vojvodić, Inverse spectral problems for Sturm-Liouville operators with a constant delay less than half the length of the interval and Robin boundary conditions, Results Math. 74 (2019), no. 1, Paper No. 45, 13. MR 3905361, DOI 10.1007/s00025-019-0972-4
Feng Wang and Chuan-Fu Yang, Traces for Sturm-Liouville operators with constant delays on a star graph, Results Math. 76 (2021), no. 4, Paper No. 220, 17. MR 4328473, DOI 10.1007/s00025-021-01529-9
Feng Wang and Chuan-Fu Yang, A partial inverse problem for the Sturm-Liouville operator with constant delays on a star graph, Results Math. 77 (2022), no. 5, Paper No. 192, 13. MR 4462804, DOI 10.1007/s00025-022-01710-8
F. Wang and C.-F. Yang, Inverse problems for Dirac operators with a constant delay less than half of the interval, arXiv:2305.10752 [math.SP], 2023.
Chuan-Fu Yang, Inverse nodal problems for the Sturm-Liouville operator with a constant delay, J. Differential Equations 257 (2014), no. 4, 1288–1306. MR 3210029, DOI 10.1016/j.jde.2014.05.011
V. A. Yurko, An inverse spectral problem for second order differential operators with retarded argument, Results Math. 74 (2019), Paper No. 71.