Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation

We consider the singular semiclassical initial value problem for the phase space Schrodinger equation. We approximate semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit approximation; we accordingly construct a semiclassical approximation of the phase space propagator, semiclassical wave packet propagator, which admits WKBM semiclassical states as initial data. By the semiclassical propagator we construct asymptotic solutions of the phase space Schrodinger equation, noting the connection of this construction to the initial value repsresentations for the Schrodinger equation

New formulas for Maslov's canonical operator in a neighborhood of focal points and caustics in 2D semiclassical asymptotics

We suggest a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds. We present the results in the two-dimensional case and illustrate them with examples