Estimation of the resolvent for a diatomic molecule in Born-Oppenheimer approximation

Making use of an adiabatic operator that takes several electronic states into account, we derive a Born-Oppenheimer approximation of the resolvent for a diatomic molecule. This is an improvement of a result in [KMW1]. Such a resolvent approximation is useful to obtain an adiabatic approximation of total cross-sections (see [Jec2]). The strategy we use, based on Mourre's commutator method and on a new kind of global escape function, may be carried over to control the resolvent of some matricial Schrodinger operators. In the same way, we obtain a semiclassical estimate for the resolvent of the semiclassical Dirac operator with scalar electric potential, extending a result of [Ce].