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

We consider a scaling limit of a nonlinear Schr"odinger 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 equation egin{equation*} ipd{}{t}psi^e(t)= -Delta psi^e(t) + g(arepsilon,mu,|( ho^e,psi^e(t))|^{2mu}) ( ho^e,psi^e(t)) ho^e end{equation*} where $ ho^{e} o delta_0$ 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.