Asymptotic Green functions: a generalized semiclassical approach for scattering by multiple barrier potentials

We show how quantum mechanical barrier reflection and transmission coefficients R and T can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V(x) = Sigma V-j((j))(x), where each V-(j) goes to zero for x --> +/-infinity, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual R-(j) and T-(j). The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.