Scattering amplitude for the Schrödinger equation with strong magnetic field
Journées équations aux dérivées partielles (2005), article no. 8, 17 p.

In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

DOI: 10.5802/jedp.20
Classification: 35B40,  35P25,  35J10,  35A35
Keywords: Scattering theory, Schrödinger equation, Magnetic fields
Michel, Laurent 1

1 LAGA, Université Paris 13, Villetaneuse
@article{JEDP_2005____A8_0,
     author = {Michel, Laurent},
     title = {Scattering amplitude for the {Schr\"odinger} equation with strong magnetic field},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {8},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2005},
     doi = {10.5802/jedp.20},
     mrnumber = {2352776},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.20/}
}
TY  - JOUR
AU  - Michel, Laurent
TI  - Scattering amplitude for the Schrödinger equation with strong magnetic field
JO  - Journées équations aux dérivées partielles
PY  - 2005
DA  - 2005///
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.20/
UR  - https://www.ams.org/mathscinet-getitem?mr=2352776
UR  - https://doi.org/10.5802/jedp.20
DO  - 10.5802/jedp.20
LA  - en
ID  - JEDP_2005____A8_0
ER  - 
%0 Journal Article
%A Michel, Laurent
%T Scattering amplitude for the Schrödinger equation with strong magnetic field
%J Journées équations aux dérivées partielles
%D 2005
%I Groupement de recherche 2434 du CNRS
%U https://doi.org/10.5802/jedp.20
%R 10.5802/jedp.20
%G en
%F JEDP_2005____A8_0
Michel, Laurent. Scattering amplitude for the Schrödinger equation with strong magnetic field. Journées équations aux dérivées partielles (2005), article  no. 8, 17 p. doi : 10.5802/jedp.20. http://archive.numdam.org/articles/10.5802/jedp.20/

[1] Agmon, S. Spectral properties of Schrödinger operators and scattering theory., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Volume 2 (1975), pp. 151-218 | Numdam | MR | Zbl

[2] Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Volume 45 (1978) no. 4, pp. 847-883 | MR | Zbl

[3] Bruneau, V.; Dimassi, M. Weak asymptotics of the spectral shift function in strong constant magnetic field (to appear)

[4] Dimassi, M. Développements asymptotiques de l’opérateur de Schrödinger avec champ magnétique fort, Comm. Partial Differential Equations, Volume 26 (2001) no. 3-4, pp. 595-627 | MR | Zbl

[5] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[6] Gérard, C. Semiclassical resolvent estimates for two and three-body Schrödinger operators, Comm. Partial Differential Equations, Volume 15 (1990) no. 8, pp. 1161-1178 | MR | Zbl

[7] Gérard, C.; Martinez, A. Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris Sér. I Math., Volume 306 (1988) no. 3, pp. 121-123 | MR | Zbl

[8] Isozaki, H.; Kitada, H. A remark on the microlocal resolvent estimates for two body Schrödinger operators, Publ. Res. Inst. Math. Sci., Volume 21 (1985) no. 5, pp. 889-910 | MR | Zbl

[9] Maslov, V. P.; Fedoryuk, M. V. Semi-classical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics, Reidel Publishing company, 1981 | Zbl

[10] Michel, L. Scattering amplitude for the Schrödinger equation with strong magnetic field and strong electric potential | MR | Zbl

[11] Michel, L. Scattering amplitude and scattering phase for the Schrödinger equation with strong magnetic field, J. Math. Phys., Volume 46 (2005), pp. 043514, 18 pages

[12] Mourre, E. Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., Volume 78 (1980/81) no. 3, pp. 391-408 | MR | Zbl

[13] Raikov, G. D.; Dimassi, M. Spectral asymptotics for quantum Hamiltonians in strong magnetic fields, Cubo Mat. Educ., Volume 3 (2001) no. 2, pp. 317-391 | MR | Zbl

[14] Reed, M.; Simon, B. Methods of modern mathematical physics. IV., Academic Press, New York, 1978 (Analysis of operators) | MR | Zbl

[15] Robert, D.; Tamura, H. Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits, Ann. Inst. Fourier (Grenoble), Volume 39 (1989) no. 1, pp. 155-192 | Numdam | MR | Zbl

[16] Vaĭnberg, B. R. Quasiclassical approximation in stationary scattering problems, Funkcional. Anal. i Priložen., Volume 11 (1977) no. 4, p. 6-18, 96 | MR | Zbl

[17] Wang, X. P. Barrier resonances in strong magnetic fields, Comm. Partial Differential Equations, Volume 17 (1992) no. 9-10, pp. 1539-1566 | MR | Zbl

Cited by Sources: