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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2018, Volume 22, Number 2, Pages 203–213
DOI: https://doi.org/10.14498/vsgtu1599
(Mi vsgtu1599)
 

This article is cited in 8 scientific papers (total in 8 papers)

Differential Equations and Mathematical Physics

On an inverse Regge problem for the Sturm–Liouville operator with deviating argument

M. Yu. Ignatiev

N. G. Chernyshevsky Saratov State University (National Research University), Saratov, 410012, Russian Federation
Full-text PDF (938 kB) Citations (8)
(published under the terms of the Creative Commons Attribution 4.0 International License)
References:
Abstract: Boundary value problem of the form $Ly=\rho^2 y$, $y(0)=y'(\pi)+i\rho y(\pi)=0$, 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 $q({}\cdot{})$ is assumed to be a real-valued function from $L_2(0,\pi)$ 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 $x\to\pi$. In this general case, the asymptotical expansion of the characteristic function of the boundary value problem as $\rho\to\infty$ 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.
Keywords: differential operators, deviating argument, constant delay, inverse spectral problems, Regge problem.
Funding agency Grant number
Russian Foundation for Basic Research 16-01-00015_a
17-51-53180_ГФЕН_а
Ministry of Education and Science of the Russian Federation 1.1660.2017/4.6
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).
Received: January 10, 2018
Revised: April 12, 2018
Accepted: June 11, 2018
First online: June 27, 2018
Bibliographic databases:
Document Type: Article
UDC: 517.984
MSC: 34A55, 34B24, 47E05
Language: English
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
Citation in format AMSBIB
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\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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  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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