Controlling Phase Space Caustics in the Semiclassical Coherent State Propagator

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The semiclassical formula for the quantum propagator in the coherent state representation $\langle \mathbf{z}'' | e^{-i\hat{H}T/\hbar} | \mathbf{z}'\rangle$ is not free from the problem of caustics. These are singular points along the complex classical trajectories specified by $\mathbf{z}'$, $\mathbf{z}''$ and $T$ where the usual quadratic approximation fails, leading to divergences in the semiclassical formula. In this paper we derive third order approximations for this propagator that remain finite in the vicinity of caustics. We use Maslov's method and the dual representation proposed in Phys.~Rev.~Lett.~{\bf 95}, 050405 (2005) to derive uniform, regular and transitional semiclassical approximations for coherent state propagator in systems with two degrees of freedom.