Asymptotic Approximations for the Phase Space Schrodinger Equation

We consider semi-classical time evolution for the phase space Schr\"{o}dinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the Anisotropic Gaussian Approximation, related to the Nearby Orbit Approximation, by which we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the Complex WKB Theory developed by Maslov, on the basis of a canonical system in double phase space related to the Berezin-Shubin-Marinov Hamilton-Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in $\mathbb{R}$.