Scattering amplitude for the Schrödinger equation with strong magnetic field
Michel, Laurent
Journées équations aux dérivées partielles, (2005), p. 1-17 / Harvested from Numdam

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.

Publié le : 2005-01-01
DOI : https://doi.org/10.5802/jedp.20
Classification:  35B40,  35P25,  35J10,  35A35
@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},
     year = {2005},
     pages = {1-17},
     doi = {10.5802/jedp.20},
     mrnumber = {2352776},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2005____A8_0}
}
Michel, Laurent. Scattering amplitude for the Schrödinger equation with strong magnetic field. Journées équations aux dérivées partielles,  (2005), pp. 1-17. doi : 10.5802/jedp.20. http://gdmltest.u-ga.fr/item/JEDP_2005____A8_0/

[1] Agmon, S. Spectral properties of Schrödinger operators and scattering theory., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Tome 2 (1975), pp. 151-218 | Numdam | MR 397194 | Zbl 0315.47007

[2] Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Tome 45 (1978) no. 4, pp. 847-883 | MR 518109 | Zbl 0399.35029

[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, Tome 26 (2001) no. 3-4, pp. 595-627 | MR 1735654 | Zbl 0926.35002

[5] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge (1999) | MR 1842043 | Zbl 0984.35118

[6] Gérard, C. Semiclassical resolvent estimates for two and three-body Schrödinger operators, Comm. Partial Differential Equations, Tome 15 (1990) no. 8, pp. 1161-1178 | MR 929103 | Zbl 0672.35013

[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., Tome 306 (1988) no. 3, pp. 121-123 | MR 1070240 | Zbl 0711.35095

[8] Isozaki, H.; Kitada, H. A remark on the microlocal resolvent estimates for two body Schrödinger operators, Publ. Res. Inst. Math. Sci., Tome 21 (1985) no. 5, pp. 889-910 | MR 817149 | Zbl 0611.35090

[9] Maslov, V. P.; Fedoryuk, M. V. Semi-classical approximation in quantum mechanics, Reidel Publishing company, Mathematical Physics and Applied Mathematics (1981) | Zbl 0458.58001

[10] Michel, L. Scattering amplitude for the Schrödinger equation with strong magnetic field and strong electric potential | MR 2131266 | Zbl 1067.81131

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

[12] Mourre, E. Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., Tome 78 (1980/81) no. 3, pp. 391-408 | MR 603501 | Zbl 0489.47010

[13] Raikov, G. D.; Dimassi, M. Spectral asymptotics for quantum Hamiltonians in strong magnetic fields, Cubo Mat. Educ., Tome 3 (2001) no. 2, pp. 317-391 | MR 1961594 | Zbl 1067.81536

[14] Reed, M.; Simon, B. Methods of modern mathematical physics. IV., Academic Press, New York (1978) (Analysis of operators) | MR 751959 | Zbl 0401.47001

[15] Robert, D.; Tamura, H. Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits, Ann. Inst. Fourier (Grenoble), Tome 39 (1989) no. 1, pp. 155-192 | Numdam | MR 1011982 | Zbl 0659.35026

[16] Vaĭnberg, B. R. Quasiclassical approximation in stationary scattering problems, Funkcional. Anal. i Priložen., Tome 11 (1977) no. 4, p. 6-18, 96 | MR 492960 | Zbl 0381.35022

[17] Wang, X. P. Barrier resonances in strong magnetic fields, Comm. Partial Differential Equations, Tome 17 (1992) no. 9-10, pp. 1539-1566 | MR 1187621 | Zbl 0795.35097