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arXiv · 2607.28963

Optimal convergence rates of the Klein-Gordon-Schr\"odinger system in the nonrelativistic limit

Weizhu Bao·Yong Lu·Zhiwei Zheng
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Abstract

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

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Weizhu Bao, Yong Lu, Zhiwei Zheng. 2026-07-31. Optimal convergence rates of the Klein-Gordon-Schr\"odinger system in the nonrelativistic limit. https://arxiv.org/abs/2607.28963

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