The resolvent and the scattering matrix at a maximum of the potential

We study the microlocal structure of the resolvent of the semi-classical Schrödinger operator with short range potential at an energy, which is a unique non-degenerate global maximum of the potential. We prove that is a semi-classical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. As an applications of this result we describe the structure of the scattering matrix of the Schrödinger operator at the kind of critical energy.