The level crossing problem in semi-classical analysis I. The symmetric case
[Le problème du croisement générique en analyse semi-classique I. Le cas symétrique]
Colin de Verdière, Yves1
1 Université Joseph Fourier, Institut Fourier (unité mixte CNRS-UJF 5582), BP 74, 38402 Saint-Martin d'Hères Cedex (France)
Annales de l'Institut Fourier, Colloque en l'honneur de Frédéric Pham, Tome 53 (2003) no. 4, pp. 1023-1054.

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.

DOI : 10.5802/aif.1973
Classification : 35C20, 35Q40, 35S30
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
Colin de Verdière, Yves 1

1 Université Joseph Fourier, Institut Fourier (unité mixte CNRS-UJF 5582), BP 74, 38402 Saint-Martin d'Hères Cedex (France) 
@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://archive.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://archive.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://archive.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, Colloque en l'honneur de Frédéric Pham, Tome 53 (2003) no. 4, pp. 1023-1054. doi : 10.5802/aif.1973. http://archive.numdam.org/articles/10.5802/aif.1973/

[1] M. Abramowitz; I. Stegun Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publication Inc., New York, 1970 | MR

[2] V. Arnold Singularities of Caustics and Wave Fronts, Kluwer, 1990 | MR | Zbl

[3] V. Arnold On the interior scattering of waves, defined by hyperbolic variational principles, J. Geom. Phys., Volume 5 (1988), pp. 305-315 | DOI | MR | Zbl

[4] J.E. Avron; A. Gordon Born-Oppenheimer wave function near level crossing, Phys. Rev. A, Volume 62-06254 (2000)

[5] M. Born; R. Oppenheimer Zür Quantentheorie der Molekeln, Annal. Phys., Volume 84 (1927), pp. 457-484 | DOI | JFM

[6] L. Boutet de Monvel; V. Guillemin The Spectral Theory of Toeplitz Operators, Princeton Univ. Press, 1981 | MR | Zbl

[7] P. Braam; H. Duistermaat Normal forms of real symmetric systems with multiplicity, Indag. Math., N.S., Volume 4 (1993) no. 4, pp. 407-421 | DOI | MR | Zbl

[8] Y. Colin de Verdière Singular Lagrangian manifolds and semi-classical analysis, Duke Math. J., Volume 116 (2003), pp. 263-298 | DOI | MR | Zbl

[9] Y. Colin de Verdière Spectres de Graphes, Soc. Math. France, 1998 | MR | Zbl

[10] Y. Colin de Verdière The level crossing problem in semi-classical analysis. II. The Hermitian case (2003) (Prépublication Institut Fourier no 603)

[11] Y. Colin de Verdière; M. Lombardi; J. Pollet The microlocal Landau-Zener formula, Ann. IHP (Phys. théorique), Volume 71 (1999), pp. 95-127 | Numdam | MR | Zbl

[12] Y. Colin de Verdière; B. Parisse Équilibre instable en régime semi-classique. I. Concentration microlocale, Commun. in PDE, Volume 19 (1994), pp. 1535-1563 | DOI | MR | Zbl

[13] Y. Colin de Verdière; J. Vey Le lemme de Morse isochore, Topology, Volume 18 (1979), pp. 283-293 | DOI | MR | Zbl

[14] J.-M. Combes 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] J.-M. Combes; R. Seiler Spectral properties of atomic and molecular systems, Quantum dynamics of molecules (Proc. NATO Adv. Study Inst.) (1979), pp. 435-482

[16] M. Combescure; D. Robert 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] P. Exner; A. Joye 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] C. Emmrich; A. Weinstein Geometry of the transport Equation in Multicomponent WKB Approximations, Commun. Math. Phys., Volume 176 (1996), pp. 701-711 | DOI | MR | Zbl

[19] F. Faure; B. Zhilinskii Topological Chern Indices in Molecular Spectra, Phys. Rev. Letters, Volume 85 (2000), pp. 960-963 | DOI

[20] F. Faure; B. Zhilinskii 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] C. Fermanian-Kammerer A non-commutative Landau-Zener formula (2002) (Prépublication Université de Cergy-Pontoise, 01/02, 1-34) | Zbl

[22] C. Fermanian-Kammerer 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] C. Fermanian-Kammerer; C. Lasser Wigner measures and codimension two crossings (2002) (Preprint mp-arc, 02-186) | MR | Zbl

[24] W. Flynn; R. Littlejohn 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] W. Flynn; R. Littlejohn Semi-classical theory of spin-orbit coupling, Phys. Rev. A, Volume 45 (1992), pp. 7697-7717 | DOI | MR

[26] G. Folland Harmonic Analysis on Phase Space, Princeton Univ. Press, 1989 | MR | Zbl

[27] P. Gérard; C. Fermanian-Kammerer Mesures semi-classiques et croisement de modes, Bull. Soc. Math. France, Volume 130 (2002), pp. 123-168 | Numdam | MR | Zbl

[28] P. Gérard; C. Fermanian-Kammerer A Landau-Zener formula for non-degenerated involutive codimension 3 crossings (sept. 2002) (Preprint) | MR | Zbl

[29] V. Guillemin Symplectic spinors and PDE, Géométrie Symplectique et Physique mathématique (Colloque CNRS Aix-en-Provence) (1974)

[30] G. Hagedorn Molecular Propagation through Electron Energy Level Crossings, Memoirs of the AMS, Volume 536 (1994) | MR | Zbl

[31] G. Hagedorn 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] G. Hagedorn Raising and lowering operators for semiclassical wave packets, Ann. Phys., Volume 269 (1998), pp. 77-104 | DOI | MR | Zbl

[33] G. Hagedorn; A. Joye 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] G. Hagedorn; A. Joye Molecular Propapagation through small avoided Crossings of Electron Energy Levels, Rev. Math. Phys., Volume 11 (1999), pp. 41-101 | DOI | MR | Zbl

[35] R. Halberg Localized coupling between surface- and bottom-intensified flow over topography, J. Phys. Oceanogr., Volume 27 (1997), pp. 977-999 | DOI

[36] P. Holm Generic elastic Media, Physica Scripta, Volume 44 (1992), pp. 122-127 | DOI

[37] A. Joye Proof of the Landau-Zener Formula, Asymptotic Analysis, Volume 9 (1994), pp. 209-258 | MR | Zbl

[38] N. Kaidi; M. Rouleux 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] M. Karasev; Y. Vorobjev Integral representations over isotropic submanifolds and equations of zero curvature, Adv. Math., Volume 135 (1998), pp. 220-286 | DOI | MR | Zbl

[40] M. Klein; A. Martinez; R. Seiler; X. Wang On the Born-Oppenheimer Approximation of Wave Operators in Molecular Scattering Theory, Commun. Math. Phys., Volume 143 (1992), pp. 607-639 | MR | Zbl

[41] M. Kline; I. Kay Electromagnetic theory and geometrical optics., Interscience Publishers, 1965 | MR | Zbl

[42] L. Landau Zur Theorie der Energieübertragung. II., Z. Phys. Sowjet., Volume 2 (1932), pp. 46-51 | Zbl

[42] L. Landau Collected papers, Pergamon Press, 1965

[43] L. Landau; E. Lifchitz Mécanique quantique (théorie non relativiste), Mir, Moscou, 1967 | Zbl

[44] A. Melin; J. Sjöstrand Fourier integral operators with complex valued phase functions, Lecture Notes in Math. (1975) no. 459 | DOI | MR | Zbl

[45] R. Melrose; G. Uhlmann Microlocal structure of involutive conical refraction, Duke Math. J., Volume 46 (1979), pp. 571-582 | DOI | MR | Zbl

[46] J. Moser On the generalization of a theorem of Liapounoff, Comm. Pure Appl. Math., Volume 11 (1958), pp. 257-271 | DOI | MR | Zbl

[47] E. Nelson Topics in dynamics. I: Flows, Princeton Univ. Press, 1969 | MR | Zbl

[48] P. Pettersson WKB expansions for systems of Schrödinger operators with crossing eigenvalues, Asymptotic Anal., Volume 14 (1997), pp. 1-48 | MR | Zbl

[49] J. Vanneste Mode Conversion for Rossby Waves over Topography, J. Phys. Oceanogr., Volume 31 (2001), pp. 1922-1925 | DOI

[50] J. von Neumann; E. Wigner Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys. Zeit., Volume 30 (1929), pp. 467-470 | JFM

[51] A. Weinstein Symplectic manifolds and their Lagrangian submanifolds., Adv. Math., Volume 6 (1971), pp. 329-346 | DOI | MR | Zbl

[52] C. Zener Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond., Volume 137 (1932), pp. 696-702 | DOI | Zbl

Cité par Sources :