Time reversal and unitarity in the frozen Gaussian approximation for semiclassical scattering

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Abstract
The time reversal and unitarity properties of the semiclassical frozen Gaussian approximation for the propagation of time dependent wave functions are explored. It is found that propagation backwards in time is the adjoint operation to propagation forward in time. This basic property of exact quantum propagation holds exactly in the frozen Gaussian approximation. Unitarity is also demonstrated for frozen Gaussian propagation within the stationary phase approximation. This latter result holds only as long as a prefactor is included in the frozen Gaussian approximation. This point is significant since this prefactor is neglected in many frozen Gaussian variations, and its inclusion can require additional computational effort. The time reversal and unitarity results can be combined to prove the very useful, but nonobvious result that time reversed propagation is the inverse of propagation forward in time in the frozen Gaussian approximation, under the same conditions for which the unitarity property holds.