Semiclassical scattering amplitude at the maximum of the potential

We study the scattering amplitude for Schrödinger operators at a critical energy level, which is a unique nondegenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiié, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.