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 
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/
@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},
     year = {1999},
     volume = {225},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a19/}
}
TY  - JOUR
AU  - R. G. Novikov
TI  - Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1999
SP  - 301
EP  - 318
VL  - 225
UR  - http://geodesic.mathdoc.fr/item/TM_1999_225_a19/
LA  - ru
ID  - TM_1999_225_a19
ER  - 
%0 Journal Article
%A R. G. Novikov
%T Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1999
%P 301-318
%V 225
%U http://geodesic.mathdoc.fr/item/TM_1999_225_a19/
%G ru
%F TM_1999_225_a19

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $E\to+\infty$.

[1] Faddeev L. D., Matematicheskie voprosy kvantovoi teorii rasseyaniya dlya sistemy trekh chastits, Tr. MIAN, 69, Izd-vo AN SSSR, L., 1963 | MR

[2] Faddeev L. D., “Rastuschie resheniya uravneniya Shrëdingera”, DAN SSSR, 165 (1965), 514–517 | Zbl

[3] Faddeev L. D., “Obratnaya zadacha kvantovoi teorii rasseyaniya, II”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Nov. dostizh., 3, VINITI, M., 1974, 93–180 | MR

[4] Manakov S. V., “Metod obratnoi zadachi rasseyaniya i dvumernye evolyutsionnye uravneniya”, UMN, 31:5 (1976), 245–246 | MR | Zbl

[5] Dubrovin B. A., Krichever I. M., Novikov S. P., “Uravnenie Shrëdingera v magnitnom pole i rimanovy poverkhnosti”, DAN SSSR, 229 (1976), 15–18 | MR | Zbl

[6] Veselov A. P., Novikov S. P., “Konechnozonnye dvumernye operatory Shrëdingera. Yavnye formuly i evolyutsionnye uravneniya”, DAN SSSR, 279 (1984), 20–24 | MR | Zbl

[7] Veselov A. P., Novikov S. P., “Konechnozonnye dvumernye operatory Shrëdingera. Vydelenie potentsialnykh operatorov. Veschestvennaya teoriya”, DAN SSSR, 279 (1984), 784–788 | MR | Zbl

[8] Beals R., Coifman R. R., “Multidimensional inverse scattering and nonlinear partial differential equations”, Proc. Symp. Pure Math., 43, 1985, 45–70 | MR | Zbl

[9] Grinevich P. G., Manakov S. V., “Obratnaya zadacha rasseyaniya dlya dvumernogo operatora Shrëdingera, $\bar\partial$-metod i nelineinye uravneniya”, Funktsion. analiz i ego pril., 20:2 (1986), 14–24 | MR | Zbl

[10] Novikov R. G., Khenkin G. M., “$\bar\partial$-Uravnenie v mnogomernoi obratnoi zadache rasseyaniya”, UMN, 42:3 (1987), 93–152 | MR | Zbl

[11] Eskin G., Ralston J., “The inverse back-scattering problem in three dimensions”, Communs Math. Phys., 124 (1989), 169–215 | DOI | MR | Zbl

[12] Eskin G., Ralston J., “Inverse back-scattering in two dimensions”, Communs Math. Phys., 138 (1991), 451–486 | DOI | MR | Zbl

[13] Novikov R. G., “The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator”, J. Funct. Anal., 103 (1992), 409–463 | DOI | MR | Zbl

[14] Novikov R. G., “Rapidly converging approximation in inverse quantum scattering in dimension 2”, Phys. Lett. A, 238 (1998), 73–78 | DOI | MR | Zbl