QLunch: Vincent Louatron

Speaker: Vincent Louatron

Title: Energy-level crossing problems for matrix-valued operators and applications to semiclassical resonances

Abstract: The phenomenon of molecular predissociation in quantum chemistry has been successfully studied under a mathematical point of view using the Born-Oppenheimer approximation, which states that the nuclei are much heavier than the electrons. The (square root of the) mass ratio is then set as the semiclassical parameter. The average predissociation lifetime is then obtained from the semiclassical distribution of the resonances of some matrix-valued Schrödinger operator P. Such distribution is known to be governed by the underlying classical trajectories associated to P.

In this talk, we will first specify such a distribution in the case where the classical trajectories cross at a finite number of points. In a second part, we will see that this problem is reduced to a microlocal study of P at the crossing points. This microlocal method is based on a normal form reduction of P at the crossing points and on a stationary phase argument.