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

Abstract

In this paper, we study 2 × 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.