Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schr\"odinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schr\"odinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter.