Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits
Annales de l'Institut Fourier, Volume 39 (1989) no. 1, p. 155-192

We study the semi-classical asymptotic behavior as (h0) of scattering amplitudes for Schrödinger operators -(1/2)h 2 Δ+V. The asymptotic formula is obtained for energies fixed in a non-trapping energy range and also is applied to study the low energy behavior of scattering amplitudes for a certain class of slowly decreasing repulsive potentials without spherical symmetry.

Nous étudions l’asymptotique semi-classique (h0) de l’amplitude de diffusion pour l’opérateur de Schrödinger -(1/2)h 2 Δ+V. Nous obtenons une formule asymptotique pour des niveaux d’énergie sans trajectoire captée. De plus la méthode s’applique à l’étude de l’amplitude de diffusion à basse énergie, pour une classe de potentiels répulsifs décroissants assez lentement (non nécessairement à symétrie sphérique).

@article{AIF_1989__39_1_155_0,
     author = {Robert, Didier and Tamura, H.},
     title = {Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {39},
     number = {1},
     year = {1989},
     pages = {155-192},
     doi = {10.5802/aif.1162},
     zbl = {0659.35026},
     mrnumber = {91c:35116},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_1_155_0}
}
Robert, Didier; Tamura, H. Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits. Annales de l'Institut Fourier, Volume 39 (1989) no. 1, pp. 155-192. doi : 10.5802/aif.1162. http://www.numdam.org/item/AIF_1989__39_1_155_0/

[1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Norm. Sup. Pisa, 2 (1975), 151-218. | Numdam | MR 53 #1053 | Zbl 0315.47007

[2] S. Agmon, Some new results in spectral and scattering theory of differential operators on Rn, Séminaire Goulaouic-Schwartz, École Polytechnique, 1978. | Numdam | Zbl 0406.35052

[3] S. Albeverio, F. Gesztesy and R. Hɸegh-Krohn, The low energy expansion in nonrelativistic scattering theory, Ann. Inst. Henri Poincaré, 37 (1982), 1-28. | Numdam | MR 83k:81093 | Zbl 0528.35076

[4] S. Albeverio, D. Bollé F. Gesztesy R. Hɸegh-Krohn and L. Streit, Low-energy parameters in nonrelativistic scattering theory, Ann. Phys., 148 (1983), 308-326. | MR 84j:81108 | Zbl 0542.35056

[5] V. Enss and B. Simon, Finite total cross sections in nonrelativistic quantum mechanics, Comm. Math. Phys., 76 (1980), 177-209. | MR 84k:81112 | Zbl 0471.35065

[6] V. Enss and B. Simon, Total cross sections in nonrelativistic scattering theory, Quantum Mechanics in Mathematics, Chemistry and Physics, edited by K.E. Gustafson and W. P. Reinhart, Plenum Press, 1981.

[7] C. Gérard and A. Martinez, Principe d'absorption limite pour des opérateurs de Schrödinger à longue portée, Université de Paris-Sud, preprint, 1987. | Zbl 0672.35013

[8] H. Isozaki and H. Kitada Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA, 32 (1985), 77-104. | MR 86j:35125 | Zbl 0582.35036

[9] H. Isozaki and H. Kitada, Scattering matrices for two-body Schrödinger operators, Sci. Papers College Arts Sci. Univ. Tokyo, 35 (1985), 81-107. | MR 87k:35196 | Zbl 0615.35065

[10] H. Isozaki and H. Kitada, A remark on the microlocal resolvent estimates for two-body Schrödinger operators, Publ. RIMS Kyoto Univ., 21 (1985), 889-910. | MR 87f:35193 | Zbl 0611.35090

[11] A. A. Kvitsinskii, Scattering by long-range potentials at low energies, Theoretical and Mathematical Physics, 59 (1984), 629-633.

[12] V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics, Reidel, 1981. | MR 84k:58226 | Zbl 0458.58001

[13] Yu. N. Protas, Quasiclassical asymptotics of the scattering amplitude for the scattering of a plane wave by inhomogeneities of the medium, Math. USSR Sbornik, 45 (1983), 487-506. | Zbl 0549.35101

[14] D. Robert, Autour de l'approximation Semi-classique, Birkhaüser, 1987. | MR 89g:81016 | Zbl 0621.35001

[15] D. Robert and H. Tamura, Semi-classical estimates for resolvents and asymptotics for total scattering cross-sections, Ann. Inst. Henri Poincaré, 46 (1987), 415-442. | Numdam | MR 912158 | MR 89b:81041 | Zbl 0648.35066

[16] B. R. Vaingerg, Quasi-classical approximation in stationary scattering problems, Func. Anal. Appl., 11 (1977), 247-257. | Zbl 0413.35025

[17] X. P. Wang, Time-decay of scattering solutions and resolvent estimates for semi-classical Schrödinger operators, Université de Nantes, preprint, 1986.

[18] K. Yajima, The quasi-classical limit of scattering amplitude — L2 — approach for short range potentials — Japan J. Math., 13 (1987), 77-126. | MR 914315 | MR 88i:35129 | Zbl 0648.35067

[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, 1979. | MR 529429 | MR 80m:81085 | Zbl 0405.47007

[20] R. G. Newton, Scattering Theory of Waves and Particles, 2nd édition, Springer, 1982. | MR 666397 | MR 84f:81001 | Zbl 0496.47011