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) 
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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://www.numdam.org/articles/10.5802/aif.1973/

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