Propagation of Quantum Expectations with Husimi Functions

TL;DR

This paper develops a second order Egorov type propagation theorem for quantum expectation values using Husimi functions, enabling improved semiclassical analysis and numerical validation in multi-dimensional Schrödinger equations.

Contribution

It introduces a novel second order Egorov theorem for Husimi functions, bridging Weyl and Anti-Wick quantizations, with proven transition and commutator rules and validated numerical experiments.

Findings

Validated the second order propagation theorem numerically in 2D and 6D Schrödinger equations.

Established transition and commutator rules for Weyl and Anti-Wick operators.

Demonstrated improved accuracy in quantum expectation propagation using Husimi functions.

Abstract

We analyse the dynamics of expectation values of quantum observables for the time-dependent semiclassical Schr\"odinger equation. To benefit from the positivity of Husimi functions, we switch between observables obtained from Weyl and Anti-Wick quantization. We develop and prove a second order Egorov type propagation theorem with Husimi functions by establishing transition and commutator rules for Weyl and Anti-Wick operators. We provide a discretized version of our theorem and present numerical experiments for Schr\"odinger equations in dimensions two and six that validate our results.