Convergence of the Semiclassical Approximation for Chaotic Scattering

The series that arises from the semiclassical approximation for scattering amplitudes is studied when the scattering is chaotic. It is argued that the terms of the series decay with an exponent equal to $\frac{1}{2d}$, where $d$ is the capacity dimension of one of the classical scattering functions. The result applies to one-dimensional inelastic and two-dimensional elastic scattering, and it is verified numerically for a one-dimensional model. An estimate of how rapidly the semiclassical series converges is given.