  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 \begin{equation*}
i\frac{\partial }{\partial t} \psi^\varepsilon(t)= -\Delta \psi^\varepsilon(t)
+ g(\varepsilon,\mu,|(\rho^\varepsilon,\psi^\varepsilon(t))|^{2\mu})
(\rho^\varepsilon,\psi^\varepsilon(t)) \rho^\varepsilon \end{equation*} where
$\rho^{\varepsilon} \to \delta_0$ 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.
