Remarks on inverse scattering in one dimension

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Abstract
This paper answers the following questions: (1) what are the consequences in the matrix‐Marchenko inversion scheme if a given S matrix lacks forward analyticity; and (2) in particular, does the condition known as the miracle depend on forward analyticity, and if not, what properties of S does it depend on? The answers are (1) if the input S matrix lacks forward analyticity then the output S matrix has it anyway, and (2) integrability of kRl,r is sufficient for the miracle to occur. It is also found that the matrix–Marchenko procedure simultaneously constructs the potentials for two scattering problems which differ only by the signs of their reflection coefficients.