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Vladimir A. Zolotarev

Publications and source records attributed to Vladimir A. Zolotarev.

7 recordsLinked to original sources
arXiv2025

Inverse scattering problem for a cubic string having the shape of a step

Problem for a cubic string having the shape of a step is studied. It is proved that here we have two scattering problems: the direct one and its dual. The direct problem describes scattering of the waves coming from ``$+\infty$'', and its dual, correspondingly, of the waves from ``$-\infty$''. The main systems of linear singular equations for both problems are obtained. It is shown how to unambiguously restore potential from the solutions to these equation systems.

math.CA↗
arXiv2023

Inverse scattering problem for operators with a finite dimensional non-local potential

Scattering problem for a self-adjoint integro-differential operator, which is the sum of the operator of second derivative and of a finite-dimensional self-adjoint operator, is studied. Jost solutions are found and it is shown that the scattering function has a multiplicative structure, besides, each of the multipliers is a scattering coefficient for a pair of self-adjoint operators, one of which is a one-dimensional perturbation of the other. Solution of the inverse problem is based upon the solutions to the inverse problem for every multiplier. A technique for finding parameters of the finite-dimensional perturbation via the scattering data is described.

math.CA↗
arXiv2023

Inverse spectral problem for a third-order differential operator

Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system of integral linear equations is obtained. Via solution to this system, the potential is calculated. It is shown that the main parameters of the obtained system of equations are expressed via spectral data of the initial operator. It is established that the potential is unambiguously defined by the four spectra.

math.CA↗
arXiv2023

Inverse scattering problem for a third-order operator with local potential

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem.

math.FA↗
arXiv2020

Direct and inverse problems for a periodic problem with non-local potential

Sum of a second derivative operator with periodic boundary conditions and an integral operator of rank one (non-local potential) is studied in this manuscript. Not only spectral analysis is conducted for this operator but the inverse problem is solved, viz., a method of the reconstruction of non-local potential by the spectral data is presented. Class of spectral data that give the solution to the inverse problem is described.

math.FA↗