The microlocal Landau-Zener formula

Yves Colin de Verdière (1) , Maurice Lombardi (2) , Joël Pollet (3)

1 IF - Institut Fourier
2 LSP - Laboratoire de Spectrométrie Physique
3 LAME-LIPhy - LAsers, Molécules et Environnement

We describe the space of microlocal solutions of a 2 \Theta 2 system of pseudo­differential operators (PDO) on the real line near an avoided cross­ ing (2­levels system). We prove Landau­Zener type formulae in the adiabatic case with avoided crossings and for the classical limit of coupled Schr¨odinger operators (Born­Oppenheimer approximation). The formulae that we get are uniform in the set of small parameters (Planck constant and coupling constant), they admits an uniquely determined complete asymptotic expansion and allow to access simply to phases which are needed in order to derive quantization conditions. The present paper is an expanded version of results already ob­ tained by Joel Pollet in his PhD thesis [22]. Quantization conditions will be described in [5], following the techniques of [8]. See also [25] concerning the scattering matrix. An extension to time dependent Schr¨odinger equation close to the work by Hagedorn [13] and Hagedorn­Joye [14] and based on [20], [10] and [27] is also in preparation.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

Yves Colin de Verdière, Maurice Lombardi, Joël Pollet. The microlocal Landau-Zener formula. Annales de l'Institut Henri Poincare Physique Theorique, 1999, 71 (1), pp.95. ⟨hal-00961343⟩