Inverse problem for Dirac operators with a small delay
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Abstract
This paper addresses inverse spectral problems associated with Dirac-type operators with a constant delay, specifically when this delay is less than one-third of the interval length. Our research focuses on eigenvalue behavior and operator recovery from spectra. We find that two spectra alone are insufficient to fully recover the potentials. Additionally, we consider the Ambarzumian-type inverse problem for Dirac-type operators with a delay. Our results have significant implications for the study of inverse problems related to the differential operators with a constant delay and may inform future research directions in this field.
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References
S. Albeverio, R. Hryniv, and Ya. V. Mykytyuk. Inverse spectral problems for Dirac
operators with summable potentials. Russ. J. Math. Phys., 12(4):406–423, 2005.
N. Bondarenko and V. Yurko. An inverse problem for Sturm-Liouville differential
operators with deviating argument. Appl. Math. Lett., 83:140–144, 2018.
S. Buterin. On the uniform stability of recovering sine-type functions with asymptotically
separated zeros. Math Notes, 111:343–355, 2022.
S. Buterin and N. Djuri´c. Inverse problems for Dirac operators with constant delay:
uniqueness, characterization, uniform stability. Lobachevskii J. Math., 43(6):1492–
, 2022.
S. A. Buterin, M. Pikula, and V. A. Yurko. Sturm-Liouville differential operators
with deviating argument. Tamkang J. Math., 48(1):61–71, 2017.
J. B. Conway. Functions of one complex variable II. Graduate texts in mathematics.
Springer, New York, NY, 1 edition, 1996.
N. Djuri´c and S. Buterin. On an open question in recovering Sturm-Liouville-type
operators with delay. Appl. Math. Lett., 113:No 106862, 2021.
N. Djuri´c and S. Buterin. On non-uniqueness of recovering Sturm-Liouville operators
with delay. Commun. Nonlinear Sci. Numer. Simul., 102:No 105900, 2021.
N. Djuri´c and S. Buterin. Iso-bispectral potentials for Sturm-Liouville-type operators
with small delay. Nonlinear Anal., Real World Appl., 63:No 103390, 2022.
N. Djuri´c and B. Vojvodi´c. Inverse problem for Dirac operators with a constant delay
less than half the length of the interval. Appl. Anal. Discrete Math., 17:249–261, 2023.
J. Eckhardt, A. Kostenko, and G. Teschl. Inverse uniqueness results for onedimensional
weighted Dirac operators. In Spectral theory and differential equations:
V. A. Marchenko’s 90th anniversary collection, pages 117–133. Providence, RI: American
Mathematical Society (AMS), 2014.
T. Erneux. Applied delay differential equations. Springer, Berlin, 2009.
G. Freiling and V. A. Yurko. Inverse problems for Sturm-Liouville differential operators
with a constant delay. Appl. Math. Lett., 25(11):1999–2004, 2012.
M. G. Gasymov and T. T. Dˇzabiev. Solution of the inverse problem by two spectra
for the Dirac equation on a finite interval. Akad. Nauk Azerba˘ıdˇzan. SSR Dokl.,
(7):3–6, 1966.
S. B. Norkin. Differential equations of the second order with retarded argument. Some
problems of the theory of vibrations of systems with retardation. Translated from
the Russian by L. J. Grimm and K. Schmitt, volume 31 of Transl. Math. Monogr.
American Mathematical Society (AMS), Providence, RI, 1972.
M. Pikula. Determination of a Sturm-Liouville-type differential operator with retarded
argument from two spectra. Mat. Vesnik, 43(3-4):159–171, 1991.
M. Pikula, V. Vladiˇci´c, and B. Vojvodi´c. Inverse spectral problems for Sturm-
Liouville operators with a constant delay less than half the length of the interval
and Robin boundary conditions. Results in Mathematics, 74(1):No 45, 2019.
A. D. Polyanin, V. G. Sorokin, and A. I. Zhurov. Delay ordinary and partial differential
equations. Adv. Appl. Math. (Boca Raton). Boca Raton, FL: CRC Press,
H. L. Smith. An introduction to delay differential equations with applications to the
life sciences, volume 57. springer New York, 2011.
V. Vladiˇci´c, M. Boˇskovi´c, and B. Vojvodi´c. Inverse problems for Sturm-Liouvilletype
differential equation with a constant delay under Dirichlet/polynomial boundary
bonditions. Bulletin of the Iranian Mathematical Society, 48(4):1829–1843, 2022.
B. Vojvodi´c, N. Djuri´c, and V. Vladiˇci´c. On recovering Dirac operators with two
delays. Complex Analysis and Operator Theory, 18(5):No 105, 2023.
B. Vojvodi´c, N. Pavlovi´c Komazec, and F.A. ¸ Cetinkaya. Recovering differential operators
with two retarded arguments. Bolet´ın de la Sociedad Matem´atica Mexicana,
(3):No 68, 2022.
B. Vojvodic and V. Vladicic. Recovering differential operators with two constant
delays under Dirichlet/Neumann boundary conditions. Journal of Inverse and Illposed
Problems, 28(2):237–241, 2020.
B. Vojvodic and V. Vladicic. On recovering Sturm–Liouville operators with two
delays. Journal of Inverse and Ill-posed Problems, 2024. doi.org/10.1515/jiip-2023-
B. Vojvodi´c, V. Vladiˇci´c, and N. Djuri´c. Inverse problem for Dirac operators with
two constant delays. Journal of Inverse and Ill-posed Problems, 32(3):573–586, 2024.
F. Wang and C.-F. Yang. Inverse problems for Dirac operators with a constant delay
less than half of the interval. J. Math. Phys., 65(3):No 032706, 2024.
F. Wang, C.-F. Yang, S. Buterin, and N. Djuri´c. Inverse spectral problems for Dirac-type
operators with global delay on a star graph. Anal. Math. Phys., 14(2):No 24,