Abstract
The topic of this paper are non-self-adjoint second-order differential operators with a constant delay, which is less than half of the length of the interval. We consider the case when a delay is from \(\tau \in [\frac{2\pi }{5},\frac{\pi }{2})\), and the potential is a real-valued function which satisfy \(q\in L^{2}[0,\pi ]\). The inverse spectral problem of recovering the potential from the spectra of two boundary value problems with Robin boundary conditions has been studied. We have proved that the delay and the potential are uniquely determined by two spectra of boundary spectral problem, one with boundary conditions \(y'(0)-hy(0)=0\), \(y'(\pi )+H_{1} y(\pi )=0\) and the other with boundary conditions \(y'(0)-hy(0)=0\), \(y'(\pi )+H_{2} y(\pi )=0\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambarzumjan, V.: Uber eine Frage der Eigenwerttheorie. Zeitshr. fr Physik -Bd. 53, 690–695 (1929)
Freiling, G., Yurko, V.: Inverse Sturm–Liouville Problems and Their Applications. Nova Science Publishers, Inc., Huntigton (2008)
Sat, M.: Inverse problems for Sturm–Liouville operators with boundary conditions depending on a spectral parameter. Electr. J. Differ. Equ. 2017(26), 1–7 (2017)
Yurko, V.A., Yang, C.F.: Recovering differential operators with nonlocal boundary conditions. Anal. Math. Phys. 6(4), 315–326 (2016)
Bondarenko, N., Buterin, S.: On recovering the Dirac operator with an integral delay from the spectrum. Results Math. 71(3), 1521–1529 (2017)
Buterin, S.A., Sat, M.: On the half inverse spectral problem for an integrodifferential operator. Inverse Probl. Sci. Eng. 25(10), 1508–1518 (2017)
Yang, C.-F.: Trace and inverse problem of a discontinuous Sturm–Liouville operator with retarded argument. J. Math. Anal. Appl. 395, 30–41 (2012)
Freiling, G., Yurko, V.L.: Inverse problems for Sturm–Liouville diferential operators with a constant delay. Appl. Math. Lett. 25(11), 1999–2004 (2012)
Buterin, S., Yurko, V.: An inverse spectral problem for Sturm–Liouville operators with a large constant delay. Anal. Math. Phys. (2017). https://doi.org/10.1007/s13324-017-0176-6
Vladicic, V., Pikula, M.: An inverse problems for Sturm–Liouville-type differential equation with a constant delay. Sarajevo J. Math. 12(1), 83–88 (2016)
Pikula, M.: Determination of a Stur–Liouville-type differential operator with delay argument from two spectra. Mat. Vesnik 43(34), 159–171 (1991)
Pikula, M., Vladicic, V., Markovic, O.: A solution to the inverse problem for Sturm–Liouville-type equation with a delay. Filomat 27(7), 1237–1245 (2013)
Buterin, S.A., Choque, A.E.: Rivero on inverse problem for a convolution integro-differential operator with Robin boundary conditions. Appl. Math. Lett 48, 150–155 (2015)
Hochstadt, H.: Integral Equations. Wiley, New York (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pikula, M., Vladičić, V. & Vojvodić, B. 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, 45 (2019). https://doi.org/10.1007/s00025-019-0972-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-019-0972-4
Keywords
- Differential operator with a delay
- inverse spectral problem
- Fourier coefficients
- Volterra integral equation