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The paper brieﬂy reviews formal methods and associated conditions for solving the forward and inverse Schrodinger ¨ scattering problem for a three-dimensional elastic scattering potential. These methods are based on an application of the Green’s function and are conditional upon the properties of the scattering potential, e.g. that the scattering potential is a ‘weak scatterer’. In this paper, we explore an alternative route to solving the problem which depends on properties imposed on the scattered waveﬁeld rather than the scattering potential. In particular, we explore the case when the gradient of the scattered waveﬁeld is weak relative to its frequency. An inverse scattering solution is then derived from which iterative forward scattering solutions can be formulated. The properties of this solution are studied including various simpliﬁcations that can be made and the conditions upon which they rely. This includes a phase only condition that is used to compute the Rutherford scattering crosssection with a second order correction. Finally, it is shown how the approach can be applied to the relativistic case when the scattering problem is determined by the Klein-Gordon equation and for electromagnetic scattering problems that are based on the inhomogeneous Helmholtz equation.
Blackledge, J.:Weak Gradient Inverse Scattering Solutions. ISAST Trans. on Computing and Intelligent Systems, vol: 3, issue: 1, pages: 15 - 23, 2011.(ISSN 1798-2448)