The level crossing problem in semi-classical analysis. II. The Hermitian case

Abstract

Our goal is to recover and extend the difficult results of George Hagedorn (1994) on the propagation of coherent states in the Born-Oppenheimer approximation in the case of generic crossings of eigenvalues of the (matrix valued) classical Hamiltonian. This problem, going back to Landau and Zener in the thirties, is often called the “Mode Conversion Problem” by physicists and occurs in many domains of physics (see the paper [23] by W. Flynn and R. Littlejohn). We want to obtain a geometrical description of the propagation of states in the framework of semi-classical analysis andWKB-Lagrangian states. It turns out that, in the very beautiful (but not well known!) paper [7] published in 1993, Peter Braam and Hans Duistermaat found that there is a formal normal form for this problem. A formal normal form for the dispersion relation were already founded by Arnold [3]. In our paper, we show, using Nelson’s wave operators method, that, in the hyperbolic case, their normal form can be extended to a local normal form in the phase space. Then, we extend this classical local normal form to the complete symbol, getting a microlocal normal form, and derive from it a precise geometric de- scription of the semi-classical propagation of states of a symmetric system of pseudo-differential equations near a generic hyperbolic codimension 3 singu- larity of the characteristic set (defined by the so called “dispersion relation”). We describe in a sketchy way the elliptic case. The complex Hermitian case will be worked out in [10].