Uniform semiclassical approximations for umbilic bifurcation catastrophes

Abstract

Gutzwiller's trace formula for the semiclassical density of states diverges at the bifurcation points of periodic orbits and has to be replaced with uniform semiclassical approximations. We present a method to derive these expressions from the standard representations of the elementary catastrophes and to directly relate the uniform solutions to classical periodic orbit parameters, thereby circumventing the numerical application of normal form theory. The technique allows an easy handling of ungeneric bifurcations with corank 2 such as the umbilic catastrophes and is demonstrated on a hyperbolic umbilic in the diamagnetic Kepler problem.