Molecular predissociation resonances near an energy-level crossing I: Elliptic interaction

We study the resonances of a two-by-two semiclassical system of one dimensional Schrödinger operators, near an energy where the two potentials intersect transversally, one of them being bonding, and the other one anti-bonding. We locate the resonances and obtain estimates on their widths, that become optimal under an additional condition of ellipticity on the interaction. Our method relies on the construction of fundamental solutions for the two scalar unperturbed operators, and on an iterative procedure in order to obtain solutions to the system.