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

We show how quantum mechanical barrier reflection and transmission coeffici ents R and T can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) poten tial. 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 f unctions by a generalized semiclassical approximation, which is based on th e usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual R-(j) an d T-(j). The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflec tion probabilities of these potentials which are very accurate when compare d with exact numerical calculations, being much better than the usual WKB a pproximation. Finally we briefly discuss how to extend the present method t o other kinds of potential.