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

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).

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.

@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},
     pages = {155--192},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {1},
     year = {1989},
     doi = {10.5802/aif.1162},
     mrnumber = {91c:35116},
     zbl = {0659.35026},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1162/}
}
TY  - JOUR
AU  - Robert, Didier
AU  - Tamura, H.
TI  - Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits
JO  - Annales de l'Institut Fourier
PY  - 1989
SP  - 155
EP  - 192
VL  - 39
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1162/
DO  - 10.5802/aif.1162
LA  - en
ID  - AIF_1989__39_1_155_0
ER  - 
%0 Journal Article
%A Robert, Didier
%A Tamura, H.
%T Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits
%J Annales de l'Institut Fourier
%D 1989
%P 155-192
%V 39
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1162/
%R 10.5802/aif.1162
%G en
%F 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, Tome 39 (1989) no. 1, pp. 155-192. doi : 10.5802/aif.1162. http://www.numdam.org/articles/10.5802/aif.1162/

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

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

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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. | EuDML | Numdam | MR | Zbl

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

[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 | Zbl

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

[20] R. G. Newton, Scattering Theory of Waves and Particles, 2nd édition, Springer, 1982. | MR | Zbl

Cité par Sources :