Semiclassical approximations to the Schrödinger equation remain important in a large variety of contexts. They play the dual role of computational tools (when exact calculations are too difficult or unnecessary) and sources of insight and intuition, even if numerical solutions are available. However, classical chaos often spoils the utility of semiclassical methods. Gutzwiller [1] gave a formal connection between periodic orbits (embedded in chaos) and eigenvalues (the trace formula). Although the trace formula is not a practical tool and even divergent, it has been the guiding light in the search for more servicable approaches. A large effort to “quantize chaos,” over many years, has begun to come to fruition. Recent progress has been dramatic, in both the time domain [2-4] and the energy domain [5-8]. Historically, however, the great bulk of the effort in semiclassical methods has taken place in the energy representation.

In 1928, Van Vleck [9] gave the time-dependent coordinate space propagator which was later modified by Gutzwiller to extend beyond caustics [1].