Wave Packet Parametrices for Evolutions Governed by Pdo’s with Rough Symbols

where a(t, x, ξ) and b(t, x, ξ) are real symbols which are continuous in t and smooth with respect to x and ξ. The operator a(t, x,D) is selfadjoint; if b = 0 then this formally guarantees that the above evolution is L well-posed and the corresponding evolution operators S(t, s) are L isometries. The b term rougly contributes to the growth or decay of energy along the flow, depending on whether b is negative or positive. We are interested in the phase space localization properties of the evolution operators S(t, s). These are best described in terms of the Bargman transform,