Semi-classical analysis of a conjoint crossing of three symmetric modes

Therefore, the potential V is subquadratic and the operator − ε 2 ∆q+V (q) is essentially self–adjoint on the Schwartz space. Such systems naturally arise in quantum chemistry when studying the dynamics of large molecules in the frame of Born-Oppenheimer approximation. The parameter ε is small because it encounters for the ratio of the mass of the electron and of the averaged mass of the nuclei (see [1] or [19] for the derivation of such equations). The solution ψ(t) does not have any direct physical interpretation but only quadratic functions of it: for example, for scalar equations, the position density |ψε(q, t)|2 gives the probability of finding the nuclei in the configuration q ∈ R at time t. Therefore, one is interested in the asymptotic description for the time evolution of the matrix-valued Wigner function of ψ(q, t),