Geodesic
Sufficient conditions for the solvability of the inverse problem of scattering theory on the entire line
Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 323-329

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Sufficient conditions which are close to necessary are given in terms of only the coefficient $r(k)=b(k)/a(k)$ for the solvability of the inverse problem of scattering theory for the Sturm–Liouville operator on the entire line. Bibliography: 4 titles. 
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     author = {B. M. Levitan},
     title = {Sufficient conditions for the solvability of the inverse problem of scattering theory on the entire line},
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B. M. Levitan. Sufficient conditions for the solvability of the inverse problem of scattering theory on the entire line. Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 323-329. http://geodesic.mathdoc.fr/item/SM_1980_36_3_a2/