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In this paper, we consider a Hamiltonian system combining a nonlinear Schr\"odinger equation (NLS) and an ordinary differential equation (ODE). This systemis a simplified model of the NLS around soliton solutions. Following Nakanishi\cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions.The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on$L^4$ in time norms.

On nonlinear profile decompositions and scattering for a NLS-ODE model