Global, uniform semiclassical approximations for quantum systems on the half-line

Motivated by recent experiments on highly excited Rydberg atoms in electrically polarized states, we consider the semiclassical analysis of quantum systems defined on the interval $0<x<\mathrm{}\infty$. Caustics plague the conventional approach, and Maslov's technique cannot be used. Instead we introduce an affine phase-space formulation that leads to a global, uniform semiclassical approximation to the wave function which fully captures the effects of caustics.