Scattering for the nonlinear Schrödinger equation with concentrated nonlinearity

We consider the cubic defocusing nonlinear Schrödinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all such solutions. Moreover, we demonstrate that the same phenomenology holds whenever nonlinear effects are sufficiently concentrated in space.