Optimal convergence rates of the Klein-Gordon-Schrödinger system in the nonrelativistic limit

Abstract

In this paper, we study the Klein-Gordon-Schrödinger system in the nonrelativistic regime $\hat{I}\mu \to 0$, where $\hat{I}\mu$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schrödinger system converges to a system of decoupled linear Schrödinger equations over a long time interval of order $\hat{I}{\mu }^{-1}$ with error estimates of the form $(1+t)\hat{I}{\mu }^2$; in particular, the error estimate for the Schrödinger component is uniform in time of the form $\hat{I}{\mu }^2$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(\hat{I}{\mu }^2)$ convergence rates coincide with the order of initial error, and thus are optimal.

25 pages