Accuracy of the semiclassical approximation for the time-dependent propagator

For chaotic systems the semiclassical approximation to the time-dependent propagator consists of a large number of terms, some of which may involve classical trajectories near focal points (caustics). Despite this the approximation has been found to remain accurate for relatively long times. In this article, that accuracy and potential reasons for its breakdown are studied. The principal tool is the Jacobi-Morse equation, the eigenvalue equation associated with the second variational derivative of the classical action. Our explanation for the accuracy lies partly in the following considerations: (1) verification of the efficacy of phase space smearing, (2) given an explicit form of the near caustic propagator, it is seen that the arguments concerning loops in phase space may be less relevant than the determination of the amount of separation in path-space, (3) and finally, the caustics in the chaotic examples studied may not be placed for maximum mischief.