Geodesic
Quasiclassical asymptotics of the scattering cross-section for the Schr\"odinger equation
Izvestiya. Mathematics , Tome 32 (1989) no. 1, pp. 141-165.

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The author considers scattering with a potential $gq(x)$, $x\in\mathbf R^m$, that decreases as $|x|\to\infty$ as a homogeneous function of degree $-\alpha$. In the domain $gk^{-1}\to\infty$, $gk^{\alpha-2}\to\infty$ the asymptotics of the forward scattering amplitude is found, as well as the total scattering cross-section averaged over a small interval of $k$. This is determined only by the behavior of $q(x)$ as $|x|\to\infty$. Dual results are obtained for strongly singular potentials. Bibliography: 16 titles. 
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D. R. Yafaev. Quasiclassical asymptotics of the scattering cross-section for the Schr\"odinger equation. Izvestiya. Mathematics , Tome 32 (1989) no. 1, pp. 141-165. http://geodesic.mathdoc.fr/item/IM2_1989_32_1_a7/

[1] Sobolev A. V., Yafaev D. R., “On the quasi-classical limit of the total scattering crosssection in nonrelativistic quantum mechanics”, Ann. Inst. H. Poincare, 44:2 (1986), 195–210 | MR | Zbl

[2] Yafaev D. R., “The eikonal approximation and the asymptotics of the total scattering cross-section for the Schrödinger equation”, Ann. Inst. H. Poincare, 44:4 (1986), 397–425 | MR | Zbl

[3] Birman M.-Sh., Yafaev D. R., “Asimptotika spektra $s$-matritsy pri potentsialnom rasseyanii”, Dokl. AN SSSR, 255:5 (1980), 1085–1087 | MR

[4] Landau L. D., Lifshits E. M., Kvantovaya mekhanika, Fizmatgiz, M., 1963

[5] Sobolev A. V., Yafaev D. R., “Fazovyi analiz v zadache rasseyaniya na radialnom potentsiale”, Zap. nauchn. sem. LOMI, 147, 1985, 155–178 | MR | Zbl

[6] Yafaev D. R., “O rezonansnom rasseyanii na otritsatelnom potentsiale”, Zap. nauchn. sem. LOMI, 138, 1984, 184–193 | MR | Zbl

[7] Amrein W., Pearson O., “A time-dependent approach to the total scattering cross section”, J. Phys. A, 12:4 (1979), 1469–1492 | DOI | MR | Zbl

[8] Enss V., Simon B., “Finite total cross-sections in nonrelativistic quantum mechanics”, Commun. Math. Phys., 76 (1980), 177–209 | DOI | MR | Zbl

[9] Birman M. Sh., Krein M. G., “K teorii volnovykh operatorov i operatorov rasseyaniya”, Dokl. AN SSSR, 144:3 (1962), 475–478 | MR | Zbl

[10] Yafaev D. R., “The low energy scattering for slowly decreasing potentials”, Commun. Math. Phys., 85 (1982), 177–196 | DOI | MR | Zbl

[11] Kvitsinskii A. A., “Rasseyanie na dalnodeistvuyuschikh potentsialakh pri malykh energiyakh”, Teor. i matem. fizika, 59:3 (1984), 472–478 | MR

[12] Vainberg B. R., “Kvaziklassicheskoe priblizhenie v statsionarnykh zadachakh rasseyaniya”, Funkts. analiz, 11:4 (1977), 6–18 | MR | Zbl

[13] Yajima K., The quasi-classical limit of scattering amplitude. I: Finite range potentials, Preprint, Tokyo, 1984 | MR

[14] Yajima K., The quasi-classical limit of scattering amplitude. $L^2$-approach for short range potentials, Preprint, Wien, 1984

[15] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965

[16] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, t. 3, Mir, M., 1982