Geodesic
Approximate Inverse Quantum Scattering at Fixed Energy in Dimension~2
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 301-318

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For the Schrödinger equation in dimension 2 we reconstruct the potential $v\in W^{N,1}_{\varepsilon}(\mathbb R^2)$, $\mathbb N\ni N\ge 3$, $\varepsilon>0$ ($N$-times smooth potential) from the scattering amplitude $f$ at fixed energy $E$ up to $O(E^{-(N-2)/2})$ in the uniform norm as $E\to+\infty$. 
@article{TM_1999_225_a19,
     author = {R. G. Novikov},
     title = {Approximate {Inverse} {Quantum} {Scattering} at {Fixed} {Energy} in {Dimension~2}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {301--318},
     publisher = {mathdoc},
     volume = {225},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a19/}
}
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R. G. Novikov. Approximate Inverse Quantum Scattering at Fixed Energy in Dimension~2. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 301-318. http://geodesic.mathdoc.fr/item/TM_1999_225_a19/