On semi-classical limit of nonlinear quantum scattering

TL;DR

This paper develops a scattering theory for nonlinear Schrödinger equations with external potentials in the semi-classical limit, connecting quantum and classical scattering behaviors and revealing a large time decoupling phenomenon for multiple initial states.

Contribution

It establishes a comprehensive scattering framework for nonlinear quantum systems in the semi-classical regime and compares quantum and classical scattering operators with uniform error estimates.

Findings

Quantum scattering theory shows potential and nonlinearity are negligible at large times.

Scattering theory for the envelope of coherent states is derived and compared to classical scattering.

Large time decoupling occurs for multiple initial coherent states.

Abstract

We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering the-ory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.