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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, Volume 21, Issue 3, Pages 282–293
DOI: https://doi.org/10.18500/1816-9791-2021-21-3-282-293
(Mi isu894)
 

This article is cited in 2 scientific papers (total in 2 papers)

Scientific Part
Mathematics

Reconstruction formula for differential systems with a singularity

M. Yu. Ignatiev

Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia
Full-text PDF (247 kB) Citations (2)
References:
Abstract: Our studies concern some aspects of scattering theory of the singular differential systems $y'-x^{-1}Ay-q(x)y=\rho By$, $x>0$ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl-type solutions of the considered differential system. Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation, provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula, we establish first the asymptotical expansions for the Weyl-type solutions as $\rho\to\infty$ with $o\left(\rho^{-1}\right)$ rate remainder estimate.
Key words: differential systems, singularity, integral equations, asymptotical expansions.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00102
20-31-70005
This work was supported by the Russian Foundation for Basic Research (projects Nos. 19-01-00102, 20-31-70005).
Received: 20.12.2020
Accepted: 22.01.2021
Bibliographic databases:
Document Type: Article
UDC: 517.984
Language: English
Citation: M. Yu. Ignatiev, “Reconstruction formula for differential systems with a singularity”, Izv. Saratov Univ. Math. Mech. Inform., 21:3 (2021), 282–293
Citation in format AMSBIB
\Bibitem{Ign21}
\by M.~Yu.~Ignatiev
\paper Reconstruction formula for differential systems with a singularity
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2021
\vol 21
\issue 3
\pages 282--293
\mathnet{http://mi.mathnet.ru/isu894}
\crossref{https://doi.org/10.18500/1816-9791-2021-21-3-282-293}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000692198400001}
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  • https://www.mathnet.ru/eng/isu/v21/i3/p282
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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