A new proof of scattering theory for the 3d radial nls with combined terms

In this paper, we give a simple proof of scattering result for the Schr\"odinger equation with combined term $i\pa_tu+\Delta u=|u|^2u-|u|^4u$ in dimension three, that avoids the concentrate compactness method. The main new ingredient is to extend the scattering criterion to energy-critical.