Remark on Asymptotic Completeness for Nonlinear Klein-Gordon Equations with Metastable States

Xinliang An, Avy Soffer

Published 2015-10-22Version 1

In this paper, we are in a further exploration of metastable states constructed by Soffer and Weinstein in [15]. These metastable states are the outcome of instabilities of excited states for Klein-Gordon equations under nonlinear Fermi golden rule. In [15], Soffer and Weinstein derived an upper bound ($1/(1+t)^{1/4}$) for an anomalously slow decay rate of these states. We derive the asymptotic behavior (including a lower bound $1/(1+t)^{1/4}$) and prove asymptotic completeness for current cases. This means that at very late stage ($t$ goes to $+\infty$) the solutions to this nonlinear Klein-Gordon equation could be written as the sum of free waves and error terms; and the $L^2_x$ norm of error terms will vanish eventually ($t$ goes to $+\infty$). Therefore, even though there is a lower bound for the anomalously slow decay rate for these states, asymptotic completeness still holds.