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We consider a scaling limit of a nonlinear Schr\"odinger equation (NLS) witha nonlocal nonlinearity showing that it reproduces in the limit of cutoffremoval a NLS equation with nonlinearity concentrated at a point. Theregularized 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 thenonlinearity and the scaling and has to be fine tuned in order to have anontrivial limit dynamics. The limit dynamics is a nonlinear version of pointinteraction in dimension three and it has been previously studied in severalpapers as regards the well-posedness, blow-up and asymptotic properties ofsolutions. Our result is the first justification of the model as the pointlimit of a regularized dynamics.

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