Scattering below the ground state for the 2d radial nonlinear Schr\"odinger equation

We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schr\"odinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.