Abstract:
In the present paper, we are concerned with the Sturm–Liouville operator L[q]u:=−u″+q(x)u subject to the separated boundary conditions. We suppose that q∈L2(0,π) and study a so-called inverse optimization spectral problem: given a potential q0 and a value λk, where k=1,2,…, find a potential ˆq closest to q0 in the norm of L2(0,π) such that the value λk coincides with k-th eigenvalue λk(ˆq) of the operator L[ˆq].
In the main result, we prove that this problem is related to the existence of a solution to a boundary value problem for the nonlinear equation
−u″+q0(x)u=λku+σu3
with σ=1 or σ=−1. This implies that the minimizing solution of the inverse optimization spectral problem can be obtained by solving the corresponding nonlinear boundary value problem. On the other hand, this relationship allows us to establish an explicit formula for the solution to the nonlinear equation by finding the minimizer of the corresponding inverse optimization spectral problem. As a consequence of this result, a new method of proving the generalized Sturm nodal theorem for the nonlinear boundary value problems is obtained.
Keywords:Sturm–Liouville operator, inverse optimization spectral problem, nodal theorem for the nonlinear boundary value problems.
Citation:
Ya. Il'yasov, N. Valeev, “On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems”, Ufa Math. J., 10:4 (2018), 122–128
\Bibitem{IlyVal18}
\by Ya.~Il'yasov, N.~Valeev
\paper On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 122--128
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\crossref{https://doi.org/10.13108/2018-10-4-122}
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Linking options:
https://www.mathnet.ru/eng/ufa454
https://doi.org/10.13108/2018-10-4-122
https://www.mathnet.ru/eng/ufa/v10/i4/p123
This publication is cited in the following 5 articles:
Yifei Jia, Jiangang Qi, Jing Li, “Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions”, Qual. Theory Dyn. Syst., 22:2 (2023)
Ya. Ilyasov, N. Valeev, “Recovery of the nearest potential field from the m observed eigenvalues”, Physica D, 426 (2021), 132985
J. Qi, B. Xie, “Extremum estimates of the L1-norm of weights for eigenvalue problems of vibrating string equations based on critical equations”, Discrete Contin. Dyn. Syst.-Ser. B, 26:7 (2021), 3505–3516
N. F. Valeev, Y. Sh. Ilyasov, “Inverse spectral problem for Sturm–Liouville operator with prescribed partial trace”, Ufa Math. J., 12:4 (2020), 19–29
Shuyuan Guo, Guixin Xu, Meirong Zhang, “On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials”, Arch. Math., 113:3 (2019), 301