Analyse à deux échelles d'une suite bornée de L2 sur une sous-variété du cotangent

In this Note, we investigate the generalization of two-scale Wigner measure to the case of submanifolds more general than symplectic and involutive ones for which they have been defined. We study the concentration of a bounded family in $L^2({\mathbb{R}}^d)$ on a submanifold of the cotangent space $T^1{\mathbb{R}}^d$ for which the restriction of the symplectic form to its tangent space is of constant rank. To cite this article: C. Fermanian Kammerer, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Semi-Classical Analysis of a Dirac Equation Without Adiabatic Decoupling

We study adiabatic decoupling for Dirac equation with some scaling which yields that the mass appears with a coefficient where is the semi-classical parameter and > 0. Therefore, the system presents an avoided crossing. The scale = 1/2 is critical: adiabatic decoupling holds for (0,1/2) while for 1/2, there is energy transfer at leading order between the two modes. We describe this transfer in terms of two-scale Wigner measures by means of Landau-Zener formula which takes into account the change of polarization of the measures after the crossing.     

A Landau-Zener Formula for Non-Degenerated Involutive Codimension 3 Crossings

In this paper, we study 2 x 2 systems of semi-classical pseudo-differential evolution equations which display eigenvalue crossing. We assume that the classical trajectories which hit the crossing set form the union of two involutive manifolds, with some non-degeneracy hypothesis, which covers the case of the Schrödinger equation with a matrix potential displaying a generic codimension 3 crossing. We derive an explicit formula for the energy transfer induced by the crossing, using two-scale Wigner measures. The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau-Zener formula. Résumé. Cet article est consacré à létude dun système de deux équations dévolution pseudo-différentielles semi-classiques présentant un croisement de modes. Nous supposons que lensemble des trajectoires classiques qui rencontrent le croisement est lunion de deux sous-variétés involutives vérifiant une hypothèse de non-dégénérescence; cette situation contient le cas de léquation de Schrödinger avec un potentiel matriciel présentant un croisement générique de codimension 3. Nous obtenons une formule explicite décrivant le transfert dénergie au-dessus du croisement, à laide des mesures de Wigner à deux échelles. La démonstration repose sur un théorème de forme normale qui permet de se ramener à une formule de Landau-Zener à valeurs opérateurs. Communicated by Bernard Helffer submitted 16/09/02, accepted: 11/03/03     

A non‐commutative Landau‐Zener formula

We study semi‐classical measures of families of solutions to a 2 × 2 Dirac system with 0 mass, which presents bands crossing. We focus on constant electro‐magnetic fields. The fact that these fields are orthogonal or not leads to different geometric situations. In the first case, one reduces to some well‐understood model problem. For studying the second case, we introduce some two‐scale semi‐classical measures associated with symplectic submanifold. These measures are operator‐valued measures and the transfer of energy at the crossing is described by a non‐commutative Landau‐Zener formula for these measures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)