Asymptotics for the Phase Space Schr\"{o}dinger Equation

TL;DR

This paper develops a semi-classical phase space propagator for the Schr"{o}dinger equation using Gaussian approximations, providing asymptotic solutions for initial WKB states and illustrating the method for sub-quadratic potentials.

Contribution

It introduces a novel semi-classical propagator in phase space based on anisotropic Gaussian approximation and derives the associated canonical system for the Schr"{o}dinger equation.

Findings

Constructed a semi-classical phase space propagator using Gaussian wave packets.

Derived the canonical system in double phase space related to existing frameworks.

Provided asymptotic solutions for the Cauchy problem with initial WKB states.

Abstract

We consider semi-classical time evolution for the phase space Schr\"{o}dinger equation. We construct a semi-classical phase space propagator in terms of semi-classical wave packets by the Anisotropic Gaussian Approximation, related to the Nearby Orbit Approximation. We deduce the canonical system in double phase space, related to the Berezin-Shubin-Marinov canonical system and construct a semi-classical asymptotic solution of the Cauchy problem for the phase space Schr\"{o}dinger equation for initial WKB states. We illustrate the method for sub-quadratic potentials in $\mathbb{R}$.