[Le problème du croisement générique en analyse semi-classique I. Le cas symétrique]
Nous décrivons une forme normale microlocale pour un système symétrique d'équations pseudo-différentielles dont le symbole principal est à valeurs matrices symétriques réelles ayant un croisement générique de valeurs propres. Nous utilisons cette forme normale pour décrire de façon précise les solutions mucrolocales.
We describe a microlocal normal form for a symmetric system of pseudo-differential equations whose principal symbol is a real symmetric matrix with a generic crossing of eigenvalues. We use it in order to give a precise description of the microlocal solutions.
Keywords: mode conversion, polarization, Born-Oppenheimer approximation, Maxwell equations, eigenvalue crossing, pseudo-differential systems, semi-classical analysis, lagrangian manifold, propagation of singularities, coherent states, symplectic spinors
Mot clés : conversion de modes, polarisation, approximation de Born-Oppenheimer, équations de Maxwell, croisements de valeurs propres, systèmes d'opérateurs pseudo-différentiels, analyse semi-classique, variétés lagrangiennes, propagation des singularités, états coh
@article{AIF_2003__53_4_1023_0,
author = {Colin de Verdi\`ere, Yves},
title = {The level crossing problem in semi-classical analysis {I.} {The} symmetric case},
journal = {Annales de l'Institut Fourier},
pages = {1023--1054},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {53},
number = {4},
year = {2003},
doi = {10.5802/aif.1973},
mrnumber = {2033509},
zbl = {02014671},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1973/}
}
TY - JOUR AU - Colin de Verdière, Yves TI - The level crossing problem in semi-classical analysis I. The symmetric case JO - Annales de l'Institut Fourier PY - 2003 SP - 1023 EP - 1054 VL - 53 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1973/ DO - 10.5802/aif.1973 LA - en ID - AIF_2003__53_4_1023_0 ER -
%0 Journal Article %A Colin de Verdière, Yves %T The level crossing problem in semi-classical analysis I. The symmetric case %J Annales de l'Institut Fourier %D 2003 %P 1023-1054 %V 53 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1973/ %R 10.5802/aif.1973 %G en %F AIF_2003__53_4_1023_0
Colin de Verdière, Yves. The level crossing problem in semi-classical analysis I. The symmetric case. Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 1023-1054. doi : 10.5802/aif.1973. http://www.numdam.org/articles/10.5802/aif.1973/
[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publication Inc., New York, 1970 | MR
[2] Singularities of Caustics and Wave Fronts, Kluwer, 1990 | MR | Zbl
[3] On the interior scattering of waves, defined by hyperbolic variational principles, J. Geom. Phys., Volume 5 (1988), pp. 305-315 | DOI | MR | Zbl
[4] Born-Oppenheimer wave function near level crossing, Phys. Rev. A, Volume 62-06254 (2000)
[5] Zür Quantentheorie der Molekeln, Annal. Phys., Volume 84 (1927), pp. 457-484 | DOI | JFM
[6] The Spectral Theory of Toeplitz Operators, Princeton Univ. Press, 1981 | MR | Zbl
[7] Normal forms of real symmetric systems with multiplicity, Indag. Math., N.S., Volume 4 (1993) no. 4, pp. 407-421 | DOI | MR | Zbl
[8] Singular Lagrangian manifolds and semi-classical analysis, Duke Math. J., Volume 116 (2003), pp. 263-298 | DOI | MR | Zbl
[9] Spectres de Graphes, Soc. Math. France, 1998 | MR | Zbl
[10] The level crossing problem in semi-classical analysis. II. The Hermitian case (2003) (Prépublication Institut Fourier no 603)
[11] The microlocal Landau-Zener formula, Ann. IHP (Phys. théorique), Volume 71 (1999), pp. 95-127 | Numdam | MR | Zbl
[12] Équilibre instable en régime semi-classique. I. Concentration microlocale, Commun. in PDE, Volume 19 (1994), pp. 1535-1563 | DOI | MR | Zbl
[13] Le lemme de Morse isochore, Topology, Volume 18 (1979), pp. 283-293 | DOI | MR | Zbl
[14] On the Born-Oppenheimer approximation, International Symposium on Mathematical Problems in Theoretical Physics, Kyoto Univ., Kyoto (Lecture Notes in Phys.), Volume 39 (1975), pp. 467-471 | Zbl
[15] Spectral properties of atomic and molecular systems, Quantum dynamics of molecules (Proc. NATO Adv. Study Inst.) (1979), pp. 435-482
[16] Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptotic Anal., Volume 14 (1997), pp. 377-404 | MR | Zbl
[17] Avoided Crossings in Mesoscopic Systems: Electron Propagation on a Non-uniform Magnetic Cylinder, J. Math. Phys., Volume 42 (2001), pp. 4707-4738 | DOI | MR | Zbl
[18] Geometry of the transport Equation in Multicomponent WKB Approximations, Commun. Math. Phys., Volume 176 (1996), pp. 701-711 | DOI | MR | Zbl
[19] Topological Chern Indices in Molecular Spectra, Phys. Rev. Letters, Volume 85 (2000), pp. 960-963 | DOI
[20] Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum, Lett. Math. Phys., Volume 55 (2001), pp. 219-239 | DOI | MR | Zbl
[21] A non-commutative Landau-Zener formula (2002) (Prépublication Université de Cergy-Pontoise, 01/02, 1-34) | Zbl
[22] Une formule de Landau-Zener pour un croisement de codimension 2, Séminaire Équations aux dérivées partielles, École Polytechnique, Volume exposé XXI, 9/04 (2002)
[23] Wigner measures and codimension two crossings (2002) (Preprint mp-arc, 02-186) | MR | Zbl
[24] Normal forms for linear mode conversion and Landau-Zener transitions in one dimension, Ann. Physics, Volume 234 (1994) no. 2, pp. 334-403 | DOI | MR
[25] Semi-classical theory of spin-orbit coupling, Phys. Rev. A, Volume 45 (1992), pp. 7697-7717 | DOI | MR
[26] Harmonic Analysis on Phase Space, Princeton Univ. Press, 1989 | MR | Zbl
[27] Mesures semi-classiques et croisement de modes, Bull. Soc. Math. France, Volume 130 (2002), pp. 123-168 | Numdam | MR | Zbl
[28] A Landau-Zener formula for non-degenerated involutive codimension 3 crossings (sept. 2002) (Preprint) | MR | Zbl
[29] Symplectic spinors and PDE, Géométrie Symplectique et Physique mathématique (Colloque CNRS Aix-en-Provence) (1974)
[30] Molecular Propagation through Electron Energy Level Crossings, Memoirs of the AMS, Volume 536 (1994) | MR | Zbl
[31] Higher Order Corrections to the Time-Dependent Born-Oppenheimer Approximation. I: Smooth Potentials, Ann. Math., Volume 124 (1986), pp. 571-590 | DOI | MR | Zbl
[32] Raising and lowering operators for semiclassical wave packets, Ann. Phys., Volume 269 (1998), pp. 77-104 | DOI | MR | Zbl
[33] Landau-Zener Transitions through small Electronic Eigenvalues Gaps in the Born-Oppenheimer Approximation, Ann. IHP (Phys. théorique), Volume 68 (1998), pp. 85-134 | Numdam | MR | Zbl
[34] Molecular Propapagation through small avoided Crossings of Electron Energy Levels, Rev. Math. Phys., Volume 11 (1999), pp. 41-101 | DOI | MR | Zbl
[35] Localized coupling between surface- and bottom-intensified flow over topography, J. Phys. Oceanogr., Volume 27 (1997), pp. 977-999 | DOI
[36] Generic elastic Media, Physica Scripta, Volume 44 (1992), pp. 122-127 | DOI
[37] Proof of the Landau-Zener Formula, Asymptotic Analysis, Volume 9 (1994), pp. 209-258 | MR | Zbl
[38] Forme normale d'un hamiltonien à deux niveaux près d'un point de branchement (limite semi-classique), C. R. Acad. Sci. Paris, Sér. I, Volume 317 (1993) no. 4, pp. 359-364 | MR | Zbl
[39] Integral representations over isotropic submanifolds and equations of zero curvature, Adv. Math., Volume 135 (1998), pp. 220-286 | DOI | MR | Zbl
[40] On the Born-Oppenheimer Approximation of Wave Operators in Molecular Scattering Theory, Commun. Math. Phys., Volume 143 (1992), pp. 607-639 | MR | Zbl
[41] Electromagnetic theory and geometrical optics., Interscience Publishers, 1965 | MR | Zbl
[42] Zur Theorie der Energieübertragung. II., Z. Phys. Sowjet., Volume 2 (1932), pp. 46-51 | Zbl
[42] Collected papers, Pergamon Press, 1965
[43] Mécanique quantique (théorie non relativiste), Mir, Moscou, 1967 | Zbl
[44] Fourier integral operators with complex valued phase functions, Lecture Notes in Math. (1975) no. 459 | DOI | MR | Zbl
[45] Microlocal structure of involutive conical refraction, Duke Math. J., Volume 46 (1979), pp. 571-582 | DOI | MR | Zbl
[46] On the generalization of a theorem of Liapounoff, Comm. Pure Appl. Math., Volume 11 (1958), pp. 257-271 | DOI | MR | Zbl
[47] Topics in dynamics. I: Flows, Princeton Univ. Press, 1969 | MR | Zbl
[48] WKB expansions for systems of Schrödinger operators with crossing eigenvalues, Asymptotic Anal., Volume 14 (1997), pp. 1-48 | MR | Zbl
[49] Mode Conversion for Rossby Waves over Topography, J. Phys. Oceanogr., Volume 31 (2001), pp. 1922-1925 | DOI
[50] Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys. Zeit., Volume 30 (1929), pp. 467-470 | JFM
[51] Symplectic manifolds and their Lagrangian submanifolds., Adv. Math., Volume 6 (1971), pp. 329-346 | DOI | MR | Zbl
[52] Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond., Volume 137 (1932), pp. 696-702 | DOI | Zbl
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