Nonrelativistic approximation in the energy space for KGS system

In this paper we study the nonrelativistic limit of the Cauchy problem for the damped and the conserved Klein-Gordon-Schrödinger (KGS) system, respectively. We prove that any finite energy solution to the damped KGS system converges to the one of Yukawa-Schrödinger (YS) system in the energy space H1⊕H1, and the solution to the conserved system goes to the corresponding one of a nonlinear Schrödinger (NLS) equation as well.