Asymptotic Theory for Quantum Bose Systems with Many Degrees of Freedom

We construct asymptotic expansions of Laplace type for the time-dependent quantum averages for Bose systems with many degrees of freedom, initially populated in coherent states. These solutions are localized in phase space, and they are different from the usual oscillating asymptotics for the quasi-classical wave functions. These expansions are valid on any fixed time interval, and caustics do not contribute to the asymptotics.