Semiclassical study of quantum scattering on the line

We study the well-known problem of 1-d quantum scattering by a potential barrier in the semiclassical limit. Using the so-called exact WKB method and semiclassical microlocal analysis techniques, we get a very precise and complete description of the scattering matrix, in particular when the energy is very close to a unique, quadratic maximum of the potential. In our one-dimensional setting, we also recover the Bohr-Sommerfeld quantization condition for the resonances generated by such a maximum.