Blow-up and scattering for the 1D NLS with point nonlinearity above the mass–energy threshold

Abstract

In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig–Merle’s compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.