Semiclassical amplitudes: Supercaustics and the whisker map

Abstract

The semiclassical approximation for a quantum amplitude is given by the sum of contributions from intersections of the appropriate manifolds in classical phase space. The intersection overlaps are just the Van Vleck determinants multiplied by a phase given by a classical action. Here we consider two nonstandard instances of this semiclassical prescription which would appear to be on shaky ground, yet the corresponding physical situations are not unusual. The first case involves momentum-space WKB theory for scattering potentials; the second is a propagator for the whisker map that arises in generic two-dimensional systems. In the former case two manifolds become asymptotically tangent, and the semiclassical formula needs to be uniformized in order to give a meaningful wave function. We give a uniformization procedure. In the latter case, there are an infinite number of intersections in phase space within a zone with the area of Planck's constant (the limit of resolution for quantum mechanics), yet the semiclassical sum over all contributions is shown to be correct.