Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials

2019 ◽  
Vol 74 (3) ◽  
pp. 373-386
Author(s):  
A. D. Agaltsov ◽  
R. G. Novikov
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data

AN INVERSE SCATTERING PROBLEM FOR NONLOCAL POTENTIALS I: THE FAMILY OF PHASE EQUIVALENT WAVEFUNCTIONS

1986 ◽  
Vol 01 (07) ◽  
pp. 449-454 ◽  
Author(s):  
V.M. MUZAFAROV
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
The Family ◽  
Consistent Approach ◽  
Nonlocal Potentials

We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.

scholarly journals Two-Step Extended Sampling Method for the Inverse Acoustic Source Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jiyu Sun ◽  
Yuhui Han
Keyword(s):  
Inverse Scattering ◽  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Indicator Function ◽  
Scattering Data ◽  
Acoustic Source ◽  
Scattering Problems ◽  
Linear Sampling Method ◽  
Ill Posed

Recently, a new method, called the extended sampling method (ESM), was proposed for the inverse scattering problems. Similar to the classical linear sampling method (LSM), the ESM is simple to implement and fast. Compared to the LSM which uses full-aperture scattering data, the ESM only uses the scattering data of one incident wave. In this paper, we generalize the ESM for the inverse acoustic source problems. We show that the indicator function of ESM, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. This behavior is similar to the ESM for the inverse scattering problem. Numerical examples are presented to show the effectiveness of the method.

The inverse scattering problem for a discrete dirac system on the whole axis

2017 ◽  
Vol 25 (6) ◽  
Author(s):  
Hidayat M. Huseynov ◽  
Agil K. Khanmamedov ◽  
Rza I. Aleskerov
Keyword(s):  
Inverse Problem ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Sufficient Condition ◽  
Dirac System ◽  
Entire Line

AbstractThis paper investigates the inverse scattering problem for a discrete Dirac system on the entire line with coefficients that stabilize to zero in one direction. We develop an algorithm for solving the inverse problem of reconstruction of coefficients. We derive a necessary and a sufficient condition on the scattering data so that the inverse problem is uniquely solvable.

scholarly journals A Regularised Total Least Squares Approach for 1D Inverse Scattering

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 216
Author(s):  
Andreas Tataris ◽  
Tristan van Leeuwen
Keyword(s):  
Integral Equation ◽  
Least Squares ◽  
Inverse Scattering ◽  
Least Squares Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Stability Estimate ◽  
Total Least Squares ◽  
Scattering Data ◽  
Least Squares Approach

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense.

Features of solving the direct and inverse scattering problems for two sets of monopole scatterers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Konstantin V. Dmitriev ◽  
Olga D. Rumyantseva
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Coefficient ◽  
Scattering Data ◽  
Scattering Problems ◽  
Scattering Coefficients ◽  
Spectral Spaces ◽  
The One ◽  
Physical Reasons

Abstract Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys 74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another.

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