×

zbMATH — the first resource for mathematics

Effet tunnel pour l’équation de Schrödinger avec champ magnétique. (Tunnel effect for the Schrödinger equation with a magnetic field). (French) Zbl 0699.35205
Les auteurs étudient la spectre “au fond du puits” pour un opérateur de Schrödinger avec champ magnétique sur \({\mathbb{R}}^ n:\) \[ P(h):=\sum^{n}_{j=1}(ih \partial_ j-A_ j)^ 2+V(x). \] Ils mettent l’accent sur les différences avec le cas sans champs magnétique (décroissance des fonctions propres, interactions) et montrent, en particulier, sur des exemples, qu’il peut y avoir diminution de l’effet tunnel entre deux fonds de puits ou bien un phénomène analogue à celui mis en evidence de manière heuristique par Aharonov et Bohm.
Reviewer: J.-C.Nosmas

MSC:
35P99 Spectral theory and eigenvalue problems for partial differential equations
35J10 Schrödinger operator, Schrödinger equation
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Y. Aharonov - D. Bohm , Significance of Electromagnetic Potentials in the Quantum Theory . The Physical Review vol. 115 ( 3 ), Aug. 1959 . MR 110458 | Zbl 0099.43102 · Zbl 0099.43102 · doi:10.1103/PhysRev.115.485
[2] J. Avron - I. Herbst - B. Simon , Schrödinger Operators with Magnetic Fields . · Zbl 0399.35029
[3] General Interactions . Duke Math. J. vol. 45 ( 4 ), Déc. 1978 .
[4] Separation of Center of Mass in Homogeneous Magnetic Fields . Ann. Physics 114 , pp. 431 - 451 ( 1978 ). MR 507741 | Zbl 0409.35027 · Zbl 0409.35027 · doi:10.1016/0003-4916(78)90276-2
[5] Atoms in Homogeneous Magnetic Fields. Comm. Math. Phys . 79 pp. 529 - 572 ( 1981 ). MR 623966 · Zbl 0464.35086
[6] J-M. Combes - R. Schrader - R. Seiler , Classical Bounds and Limits for Energy Distributions of Hamilton Operators in Electromagnetic Fields . Ann. Physics 111 , pp. 1 - 18 ( 1978 ). MR 489509
[7] B. Helffer , An introduction to Semi Classical Analysis for the Schrôdinger Operator. Cours en Chine . (A paraître Lecture Notes in Mathematics 1988 ). · Zbl 0647.35002
[8] B. Helffer - D. Robert , Calcul Fonctionnel Admissible et Applications . J. Funct. Anal. 53 ( 3 ) ( 1983 ), pp. 246 - 268 . MR 724029 | Zbl 0524.35103 · Zbl 0524.35103 · doi:10.1016/0022-1236(83)90034-4
[9] B. Helffer - J. Sjöstrand , Multiple Wells in the Semi-Classical Limit [1] Comm. in P.D.E. , 9 ( 4 ) ( 1984 ), pp. 337 - 408 . MR 740094 · Zbl 0546.35053 · doi:10.1080/03605308408820335
[10] Interaction Moléculaire, Symétries, perturbation : Ann. Inst. H. Poincaré , vol. 42 ( 2 ) ( 1985 ), pp. 127 - 212 . Numdam | MR 798695 · Zbl 0595.35031 · numdam:AIHPA_1985__42_2_127_0
[11] B. Helffer - A. Martinez - D. Robert , Ergodicité et Limite Semiclassique ( Comm. Math. Phys .). Article | Zbl 0624.58039 · Zbl 0624.58039 · doi:10.1007/BF01215225 · minidml.mathdoc.fr
[12] L. Hörmander , The Analysis of Linear Partial Differential Equations. Tome 3 , Chap. XXII - Springer . · Zbl 0191.10901
[13] W. Hunziker , Schrôdinger Operators with Electric or Magnetic Fields. Proc. Int. Conf. in Math. Phys ., Lausanne ( 1980 ). MR 582602 | Zbl 0471.47010 · Zbl 0471.47010
[14] I.V. Ivrii - Note au C.R.A.S. 1986 , 302 Sér I ( 13 ), pp. 467 - 471 . MR 838401
[15] T. Kato : [1] Israel J. Math. 13 ( 1972 ) pp. 125 - 174 . MR 333833
[16] Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften 132 . Springer Verlag . Zbl 0531.47014 · Zbl 0531.47014
[17] A. Menikoff - J. Sjöstrand . On the Eigenvalues of a class of hypoelliptic Operators . Math. Ann. 235 , pp. 55 - 85 ( 1978 ). MR 481627 | Zbl 0375.35014 · Zbl 0375.35014 · doi:10.1007/BF01421593 · eudml:163144
[18] A. Mohamed . [SI] B. Simon : [1] Indiana Univ. Math. J. 26 , pp. 1067 - 1073 ( 1977 ). MR 461209 | Zbl 0389.47021 · Zbl 0389.47021 · doi:10.1512/iumj.1977.26.26086
[19] Semi-Classical Analysis of Low Lying Eingenvalues I . Ann. Inst. H. Poincaré t. 38 ( 1983 ), pp. 295 - 307 .
[20] Semi-Classical Analysis of Low Lying Eingenvalues II. Tunneling . Ann. of Math . ( 1984 ) vol. 20 , pp. 89 - 118 . Zbl 0626.35070 · Zbl 0626.35070 · doi:10.2307/2007072
[21] J. Sjöstrand : [1] Singularités analytiques microlocales - Astérisque 95 ( 1982 ). MR 699623 | Zbl 0524.35007 · Zbl 0524.35007
[22] Analytic Wave front Sets and Operators with Multiple Characteristics Hokkaido Math. Journal Vol. XII ( 3 ), pp. 392 - 433 . Zbl 0531.35022 · Zbl 0531.35022
[23] M. Wilkinson , Narrowly avoided crossings Preprint April 1986 . · Zbl 0627.58035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.