Geodesic
Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy
Teoretičeskaâ i matematičeskaâ fizika, Tome 66 (1986) no. 2, pp. 234-240 
R. G. Novikov. Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy. Teoretičeskaâ i matematičeskaâ fizika, Tome 66 (1986) no. 2, pp. 234-240. http://geodesic.mathdoc.fr/item/TMF_1986_66_2_a7/
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     author = {R. G. Novikov},
     title = {Construction of two-dimensional {Schr\"odinger} operator with given scattering amplitude at fixed energy},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {234--240},
     year = {1986},
     volume = {66},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_66_2_a7/}
}
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The classical necessary properties of the scattering amplitude (reciprocity and unitarity) are, provided its $L_2$ norm is small, sufficient for the existence of a two-dimensional Schrödinger operator with the given scattering amplitude at fixed energy.

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