Finite-difference approximation for inverse scattering problem

For the first time potentials are reconstructed in a finite-difference approximation using a genuine inverse scattering method instead of multiple repeated solutions of a direct problem with iterative fitting of scattering data. Up to now a fundamental difference between spectral properties of the Schrödinger operator and its discrete analog hindered from doing this.     

Solution of an inverse problem in scattering theory

Hoyt (1939) and Firsov devised methods in classical mechanics to deduce a central scattering potential from a measured differential effective cross-section in the nonrelativistic case. These methods are here extended to the relativistic case. A detailed analysis of the applicability of all methods has been undertaken for potentials of the form V(r) = ±r^–k for sufficiently high energies of the colliding particles. It is found that Hoyt's method is inapplicable in the relativistic case only when the potential represents attraction. A relatively simple method is given for deducing the parameters and k for a monotonic attraction potential that can be approximated by V(r) = –r^–k. The method is based on simple arguments concerning the dimensions of the cross-section. It is sufficient to know only two values of the integral cross-section in the same range of angles but at different energies to determine the parameters.     

Supersymmetric Yang-Mills equations as an inverse scattering problem

The equations of motion (for N=3, 4) and the constraint equations (N=1, 2) for supersymmetric Yang-Mills theories are shown to be the compatibility conditions of some system of linear equations with a parameter.     

An inverse scattering problem from an impedance obstacle

In this paper we consider an inverse scattering problem from an obstacle with impedance boundary condition. Our aim is to recover the unknown scatterer from the far field pattern iteratively assuming the impedance function. Our method, while remaining in the framework of Newton’s method, based on a system of two nonlinear integral equations which is equivalent to the original inverse problem, avoids the need of calculating a direct problem at each iteration. Because of the ill-posedness of this problem, regularization method for example, Tikhonov regularization, is incorporated in our solution scheme. Several numerical examples with only one incident wave are given at the end of the paper to show the feasibility of our method.     

Three-body inverse scattering problem

Two methods are suggested to reconstruct three-body potentials from three-body scattering data. This was achieved by using the reduction of the corresponding Schrödinger equation to a system of ordinary differential equations (not integro-differential equations as usual in the direct problem). Exactly solvable three-body models are presented. A new simple method for solving the multi-dimensional inverse problem in a finite-difference approximation is considered in the Appendix.     

Inverse scattering problem in the relativistic quasiclassical approximation

We apply the quasipotential approach of quantum field theory to solve the inverse scattering problem in the relativistic quasiclassical approximation. We obtain expressions for reconstructing the quasipotential from the phase shifts and consider both nonrelativistic and ultrarelativistic cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No-5, pp. 11–17, May, 1987.The authors are deeply grateful to Yu. S. Vernov and N. B. Skachkov for their interest and helpful discussion of the results.     

Imaging of random surfaces and inverse scattering in the Kirchoff approximation

Abstract

Imaging of random surfaces can be modelled by integration over angular spectra of scattered plane waves. This approach suggests the representation of surface scattering in the Kirchhoff approximation using the concept of three-dimensional spatial frequencies. Optical methods of surface profiling can thus be modelled, leading to an insight into reconstruction of surface profiles from scattering data. The methods can also be extended to cover thin-film multilayer structures.     

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations are shown to be conditions of compatibility of two linear equations. All the N-instanton fields are constructed explicitly.     

Numerical solution for an inverse scattering problem of non-periodic rough surfaces

The paper presents the first results of a numerical study of inverse diffraction devoted to non-periodic rough surfaces in optics. Two kinds of rough surfaces are considered: first gratings with a finite number of grooves, and second random rough surfaces. For shallow surfaces, adequate Fourier theories have been employed with success. On the other hand, for deeper asperities, rigorous methods are needed and generally, the reconstruction of the profile is more difficult. For both Fourier and rigorous methods, the limit of resolution is studied numerically and numerous examples of reconstruction are given.Instituto Politecnico National, Escuela Superior de Fisica y Matematicas, Mexico, D.F., Mexico     

A second degree Newton method for an inverse obstacle scattering problem

A regularized second degree Newton method is proposed and implemented for the inverse problem for scattering of time-harmonic acoustic waves from a sound-soft obstacle. It combines ideas due to Johansson and Sleeman [18] and Hettlich and Rundell [8] and reconstructs the obstacle from the far field pattern for scattering of one incident plane wave.     

Uniqueness of inverse scattering problem in local quantum physics

It is shown that the a Bisognano-Wichmann-Unruh inspired formulation of local quantum physics which starts from wedge-localized algebras, leads to a uniqueness proof for the scattering problem. The important mathematical tool is the thermal KMS aspect of localization and its strengthening by the requirement of crossing symmetry for generalized formfactors.     

A method to solve an acoustic inverse scattering problem involving smart obstacles

A time-harmonic acoustic inverse scattering problem involving smart obstacles is formulated and a method to solve it is proposed. A smart obstacle is an obstacle that, when hit by an incoming acoustic wave, tries to pursue a given goal circulating a suitable pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals pursued by the smart obstacles that we have considered are the following ones: to be undetectable or to appear with a shape and/or acoustic boundary impedance different from its actual ones eventually in a location in space different from the actual location. The following time-harmonic inverse scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time-harmonic waves, the knowledge of the goal pursued by the smart obstacle and of its acoustic boundary impedance reconstruct the boundary of the obstacle. A method to solve this inverse problem that generalizes the so-called Herglotz function method is proposed. Some numerical experiments that validate the method proposed are presented. The website http://www.econ.univpm.it/recchioni/w13 contains some auxiliary material that helps the understanding of the current paper.     

Quadratic spherical bessel functions and the inverse scattering problem

In the course of inverting the partial-wave Born approximation, a new expression for the inverse function ofj _l ² (ρ) was obtained. Using this result, one can also derive two expressions involving the binomial coefficients. Finally, a particular differential operator whose effect onj _l ² (ρ) was previously investigated by Mavromatis and Jalal is shown to have similar effects onn _l ² (ρ) andn _l (ρ)j _l (ρ).     

Relevance vector machine technique for the inverse scattering problem

A novel method based on the relevance vector machine(RVM) for the inverse scattering problem is presented in this paper.The nonlinearity and the ill-posedness inherent in this problem are simultaneously considered.The nonlinearity is embodied in the relation between the scattered field and the target property,which can be obtained through the RVM training process.Besides,rather than utilizing regularization,the ill-posed nature of the inversion is naturally accounted for because the RVM can produce a probabilistic output.Simulation results reveal that the proposed RVM-based approach can provide comparative performances in terms of accuracy,convergence,robustness,generalization,and improved performance in terms of sparse property in comparison with the support vector machine(SVM) based approach.     

The inverse scattering problem and the equivalent local potential

We examine the role of the non-locality of the potential in the two-body scattering by taking the nucleon-Σ system as an example. We employ a non-local potential for the nucleon-Σ channel which has the characteristic features of the quark model and reproduces the phase shifts of widely used local potentials. We use inverse scattering methods to obtain the equivalent local potential from the non-local potential, and show that the obtained local potential has a strong short range repulsion.     

General solution of relativistic one-channel inverse scattering problem

We find explicitly in the p-representation the kernels of the logarithms of unitary operators transforming the free Hamiltonian into the general solution of the one-channel inverse scattering problem, when bound states are absent. Then we construct the transformation of the Hamiltonian adding to its spectrum the given set of bound states without changing the scattering operator. Other generators of the Poincaré group are constructed in a similar way. The proof of some relevant limits is given.     

The inverse scattering problem for singular oscillating potentials

In this paper we show how to apply the Gel'fand-Levitan equations with singular oscillating potentials. We give the general procedure for solving these equations and we illustrate the method by some examples.