Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Ignatyev M. Y. Reconstruction formula for differential systems with a singularity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 282-293. DOI: 10.18500/1816-9791-2021-21-3-282-293, EDN: CGDNRR

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Reconstruction formula for differential systems with a singularity

Ignatyev M. Yu., Saratov State University

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\linebreak  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. For\-mu\-las 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.

This work was supported by the Russian Foundation for Basic Research (projects Nos. 19-01-00102, 20-31-70005).
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