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Inverse Problems

Inverse scattering using the Heitler equation

Anthony J. Devaney, Arthur B. Weglein

Published Dec 1, 1989• 8 citations
Publisher

Abstract

The problem of determining a scattering potential from scattered field data is addressed within the two integral equation formulation of scattering theory first proposed by Goldberger (1964). The lowest-order approximation within this formulation casts the inverse problem entirely in terms of the Heitler equation which relates the on-shell T matrix directly to the scattering potential. This lowest-order approximation is applied to the case of non-relativistic quantum mechanical inverse scattering from one-dimensional local potentials and is found to yield an exact solution for the case of a one-dimensional delta function potential.

DOI
10.1088/0266-5611/5/6/001

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The problem of determining a scattering potential from scattered field data is addressed within the two integral equation formulation of scattering theory first proposed by Goldberger (1964). The lowest-order approximation within this formulation casts the inverse problem entirely in terms of the Heitler equation which relates the on-shell T matrix directly to the scattering potential. This lowest-order approximation is applied to the case of non-relativistic quantum mechanical inverse scattering from one-dimensional local potentials and is found to yield an exact solution for the case of a one-dimensional delta function potential.
Approximating the SSS matrix for solving the Marchenko equation: the case of channels with different thresholds
This work extends previous results on the inverse scattering problem within the framework of Marchenko theory (fixed-lll inversion). In particular, I approximate an nnn-channel SSS-matrix as a function of the first-channel momentum qqq by a sum of a rational term and a truncated sinc series for each matrix element. Relativistic kinematics are taken into account through the correct momentum-energy relation, and the necessary minor generalization of Marchenko theory is given. For energies where only a subset of scattering channels is open, the analytic structure of the SSS-matrix is analyzed. I demonstrate that the submatrix corresponding to closed channels, particularly near their thresholds, can be reconstructed from the experimentally accessible submatrix of open channels.The convergence of the proposed method is verified by applying it to data generated from a direct solution of the scattering problem for a known potential, and comparing the reconstructed potential with the original one. Finally, the method is applied to the analysis of S31S_{31}S31​ πNπNπN scattering data.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.