Adiabatic perturbation theory. Semiclassical limit of the reflection coefficient on a potential barrier

Using methods of Adiabatic Perturbation Theory, the asymptotic behaviour in the semiclassical limit of the reflection coefficient for one dimensional collisions in quantum mechanics is studied rigorously. In this limit we give a recursive method for an exact calculation to all orders. In contrast to other methods (which fail), our method still works when the energy is near or equal to the top of the potential. First we derive the coefficient for a specific potential. Then we give the coefficient to first three orders for an arbitrary potential depending only on the potential's behaviour around its maximum. We point out that each term needs to be analysed to ''all orders''. (orig.)