Abstract:
Our studies concern some aspects of scattering theory of the singular differential systems y′−x−1Ay−q(x)y=ρBy, x>0 with n×n matrices A,B,q(x),x∈(0,∞), where A,B are constant and ρ is a spectral parameter. We concentrate on the important special case when q(⋅) is smooth and q(0)=0 and derive a formula that express such q(⋅) 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 ρ→∞ with o(ρ−1) rate remainder estimate.
Key words:differential systems, singularity, integral equations, asymptotical expansions.
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
\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}
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This publication is cited in the following 2 articles:
M. Yu. Ignat'ev, “Constructive solution of scattering inverse problem for systems of ordinary differential equations with singularities”, Moscow University Mathematics Bulletin, 78:2 (2023), 83–94
Xin-Jian Xu, Chuan-Fu Yang, Vjacheslav A. Yurko, Ran Zhang, “Inverse problems for radial Schrödinger operators with the missing part of eigenvalues”, Sci. China Math., 66:8 (2023), 1831