no. 6
Semiclassics of the quantum current in very strong magnetic fields
[Limite semi-classique du courant quantique en présence d'un champ magnétique très fort]
Fournais, Soren1
1 Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex (France)
Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1901-1945.

Nous donnons l’asymptotique du courant d’un gaz d’électrons en limite semi-classique dans le régime champ magnétique constant et très fort. Nous supposons très peu de régularité pour le potentiel scalaire V. En particulier, le résultat peut s’appliquer au champ moyen V MTF provenant de la théorie de Thomas-Fermi magnétique. La démonstration repose sur une estimation de la densité d’états au deuxième niveau de Landau.

We prove a formula for the current in an electron gas in a semiclassical limit corresponding to strong, constant, magnetic fields. Little regularity is assumed for the scalar potential V. In particular, the result can be applied to the mean field from magnetic Thomas-Fermi theory V MTF . The proof is based on an estimate on the density of states in the second Landau band.

DOI : 10.5802/aif.1938
Classification : 81V45, 81V70
Keywords: semiclassics, magnetic Thomas-Ferni theory, quantum current
Mot clés : limite semi-classique, théorie magnétique de Thomas-Ferni, courant quantique
Fournais, Soren 1

1 Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex (France) 
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Fournais, Soren. Semiclassics of the quantum current in very strong magnetic fields. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1901-1945. doi : 10.5802/aif.1938. http://archive.numdam.org/articles/10.5802/aif.1938/

[AdMBG91] W.O. Amrein; A.-M. Boutet; de Monvel; - Berthier; V. Georgescu Notes on the N-body Problem, Part II (1991) (Univ. of Genève Preprint)

[AHS78] J. Avron; I. Herbst; B. Simon Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J, Volume 45 (1978) no. 4, pp. 847-883 | DOI | MR | Zbl

[AHS81] J. E. Avron; I. W. Herbst; B. Simon Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field, Comm. Math. Phys, Volume 79 (1981) no. 4, pp. 529-572 | MR | Zbl

[Fou01a] S. Fournais On the semiclassical asymptotics of the current and magnetisation of a non-interacting electron gas at zero temperature in a strong constant magnetic field, Ann. Henri Poincaré (2001) no. 2, pp. 1-23 | MR | Zbl

[Fou01b] S. Fournais The magnetisation of large atoms in strong magnetic fields, Comm. Math. Phys, Volume 216 (2001) no. 2, pp. 375-393 | DOI | MR | Zbl

[Fou99] S. Fournais Semiclassics of the quantum current in a strong constant magnetic field (1999) (University of Aarhus Preprint, no. 9) | Zbl

[FW94] R. Froese; R. Waxler The spectrum of a hydrogen atom in an intense magnetic field, Rev. Math. Phys, Volume 6 (1994) no. 5, pp. 699-832 | DOI | MR | Zbl

[GG99] V. Georgescu; C. Gérard On the virial theorem in quantum mechanics, Comm. Math. Phys, Volume 208 (1999) no. 2, pp. 275-281 | DOI | MR | Zbl

[Ivr98] V. Ivrii Microlocal analysis and precise spectral asymptotics, Springer-Verlag, Berlin, 1998 | MR | Zbl

[Lie81] E. H. Lieb Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys, Volume 53 (1981) no. 4, pp. 603-641 | DOI | MR | Zbl

[LL97] Elliott H. Lieb; M. Loss Analysis, American Mathematical Society, Providence, RI, 1997 | MR | Zbl

[LSY94a] E. H. Lieb; J. P. Solovej; J. Yngvason Asymptotics of heavy atoms in high magnetic fields. II. Semiclassical regions, Comm. Math. Phys, Volume 161 (1994) no. 1, pp. 77-124 | DOI | MR | Zbl

[LSY94b] E. H. Lieb; J. P. Solovej; J. Yngvason Asymptotics of heavy atoms in high magnetic fields. I. Lowest Landau band regions, Comm. Pure Appl. Math, Volume 47 (1994) no. 4, pp. 513-591 | DOI | MR | Zbl

[RS78] M. Reed; B. Simon Methods of modern mathematical physics I-IV, Academic Press, 1972-78 | MR | Zbl

[Sob94] A. V. Sobolev The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneous magnetic field, Duke Math. J, Volume 74 (1994) no. 2, pp. 319-429 | MR | Zbl

[Sob95] A. V. Sobolev Quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a moderate magnetic field, Ann. Inst. H. Poincaré Phys. Théor., Volume 62 (1995) no. 4, pp. 325-360 | Numdam | MR | Zbl

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