Chern-Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern-Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)-(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.