Abstract:
Boundary value problem of the form Ly=ρ2yLy=ρ2y, y(0)=y′(π)+iρy(π)=0, where L is the Sturm–Liouville operator with constant delay a is studied. The boundary value problem can be considered as a generalization of the classical Regge problem. The potential q(⋅) is assumed to be a real-valued function from L2(0,π) equal to 0 a.e. on (0,a). No other restrictions on the potential are imposed, in particular, we make no additional assumptions regarding an asymptotical behavior of q(x) as x→π. In this general case, the asymptotical expansion of the characteristic function of the boundary value problem as ρ→∞ contains no leading term. Therefore, no explicit asymptotics of the spectrum can be obtained using the standard methods.
We consider the inverse problem of recovering the operator from given subspectrum of the boundary value problem.
Inverse problems for differential operators with deviating argument are essentially more difficult with respect to the classical inverse problems for differential operators. “Non-local” nature of such operators is insuperable obstacle for classical methods of inverse problem theory.
We consider the inverse problem in case of delay, which is not less than the half length of the interval and establish that the specification of the subspectrum of the boundary value problem determines, under certain conditions, the potential uniquely.
Corresponding subspectra are characterized in terms of their densities. We also provide a constructive procedure for solving the inverse
problem.
This work was supported by the Russian Foundation for
Basic Research (projects nos. 16–01–00015_a, 17–51–53180_GPhEN_a) and Russian Ministry of Education and Science (project no. 1.1660.2017/4.6).
Citation:
M. Yu. Ignatiev, “On an inverse Regge problem for the Sturm–Liouville operator with deviating argument”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:2 (2018), 203–213
\Bibitem{Ign18}
\by M.~Yu.~Ignatiev
\paper On an inverse Regge problem for the Sturm--Liouville operator with deviating argument
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2018
\vol 22
\issue 2
\pages 203--213
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\crossref{https://doi.org/10.14498/vsgtu1599}
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This publication is cited in the following 8 articles:
Feng Wang, Chuan-Fu Yang, “Incomplete inverse problem for Dirac operator with constant delay”, Proc. Amer. Math. Soc., 2024
Feng Wang, Chuan-Fu Yang, “Inverse problems for Dirac operators with constant delay less than half of the interval”, Journal of Mathematical Physics, 65:3 (2024)
Nebojsa Djuric, Biljana Vojvodic, “Inverse problem for Dirac operators with a constant delay less than half the length of the interval”, Appl Anal Discrete Math, 17:1 (2023), 249
Djuric N., Buterin S., “Iso-Bispectral Potentials For Sturm-Liouville-Type Operators With Small Delay”, Nonlinear Anal.-Real World Appl., 63 (2022), 103390
F. Wang, Ch.-F. Yang, “Traces for Sturm-Liouville operators with constant delays on a star graph”, Results Math., 76:4 (2021), 220
S. A. Buterin, M. A. Malyugina, C.-T. Shieh, “An inverse spectral problem for second-order functional-differential pencils with two delays”, Appl. Math. Comput., 411 (2021), 126475
N. Djuric, S. Buterin, “On non-uniqueness of recovering Sturm-Liouville operators with delay”, Commun. Nonlinear Sci. Numer. Simul., 102 (2021), 105900
N. Djuric, S. Buterin, “On an open question in recovering Sturm-Liouville-type operators with delay”, Appl. Math. Lett., 113 (2021), 106862