Abstract
In this paper we discuss one-dimensional scattering and inverse scattering for the Helmholtz equation on the half-line from the point of view of the layer stripping. By full or nonlinear scattering, we mean the transformation between the index of refraction (actually half of its logarithmic derivative) and the reflection coefficient. We refer to this mapping as nonlinear scattering, because the mapping itself is nonlinear. Another appropriate name is multiple scattering, as this model includes the effects of multiple reflections.By linear scattering we mean the Born, or single scattering, approximation. This is the Frechet derivative of the full scattering transform at the constant index of refraction, which can be calculated to be exactly the Fourier transform. In [J. Sylvester, D. P. Winebrenner, and F. Gylys-Colwell, SIAM J. Appl. Math., 56 (1996), pp. 736--754], we introduced a variant of layer stripping based on causality and the Riesz transform, rather than on trace formulas---see [A. Brickstein and...
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