Semiclassical Propagation of Coherent States for the Hartree equation

TL;DR

This paper develops a semiclassical approximation method for the nonlinear Hartree equation with external potential, using time-dependent coherent states and classical flow, with error estimates and asymptotic expansion.

Contribution

It introduces a new semiclassical approximation framework for the Hartree equation with explicit error bounds and asymptotic expansion techniques.

Findings

Approximation error in L^2 norm is proportional to sqrt of Planck's constant.

Solution can be effectively represented by a time-dependent coherent state.

Formal asymptotic expansion of the solution is provided.

Abstract

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion.