On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles

TL;DR

This paper analyzes the long-term behavior of a quantum system with heavy and light particles, providing explicit error bounds and extending results to arbitrary interaction strengths in certain cases.

Contribution

It offers a detailed characterization of the asymptotic dynamics of a mixed heavy-light particle quantum system with small mass ratio and extends results to arbitrary interaction strength when there is a single heavy particle.

Findings

Explicit asymptotic dynamics with error control for small mass ratio

Extension of results to arbitrary interaction strength for one heavy particle

Application to decoherence effects on heavy particles

Abstract

We consider a non relativistic quantum system consisting of $K$ heavy and $N$ light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential $\alpha V$. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming $\alpha$ sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K=1 the result is extended to arbitrary $\alpha$. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schr\"{o}dinger group. Exploiting the asymptotic formula, it is also outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones.