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 
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/
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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.

[1] L. D. Faddeev, Itogi nauki, t. 3, VINITI, Moskva, 1973

[2] L. D. Faddeev, “Svoistva $S$-matritsy odnomernogo uravneniya Shredingera”, Trudy matem. in-ta im. V. A. Steklova AN SSSR, CXXIII, 1964, 314–336 | MR | Zbl

[3] F. D. Gakhov, Kraevye zadachi, Fizmatgiz, Moskva, 1958

[4] N. N. Muskhelishvili, Singulyarnye integralnye uravneniya, Gostekhizdat, Moskva, 1946