Remarks on asymptotic solutions for time dependent Schrodinger equations

For smooth Hamiltonians (C-infinity or analytic), we consider approximate solutions for the time dependent Schrodinger equation in the semi-classical regime i.e when the Planck constant h is small, starting at time t = to with a Gaussian function picked at a point z in the phase space. These solutions are defined by power series in rooth. We give here accurate estimates for the coefficients and for the remainder terms with control in the Planck constant h, the time t and the order N of truncation of the series.