Semiclassically weak reflections above analytic and non-analytic potential barriers

The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross) -1 )); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross) N . If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n ) gives r approximately exp(-h(cross) -n (n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10 -1000 .