The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equationi∂∂tψε(t)=−Δψε(t)+g(ε,μ,|(ρε,ψε(t))|2μ)(ρε,ψε(t))ρε where ρε→δ0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.