Semiclassical Representation of the Scattering Matrix by a Feynman Integral

We study the scattering problem for one-dimensional Schr?dinger equations in the semiclassical limit when the energy level is close to the quadratic maxima of the potential. Starting from the formula of the scattering matrix obtained by the exact WKB method, we represent its principal term in a reduced form of the Feynman integral, an absolutely convergent sum of suitably defined probability amplitudes over countably many trajectories on ℝ generated by classical trajectories and tunneling effects. Received: 14 November 1997 / Accepted: 6 April 1998