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 $i{\text{pa}}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{2}u-\mathrm{\mid }u{\mathrm{\mid }}^{4}u$ in dimension three, that avoids the concentrate compactness method. The main new ingredient is to extend the scattering criterion to energy-critical.