  In the present paper we study the following scaled nonlinear Schr\"odinger
equation (NLS) in one space dimension: \[ i\frac{d}{dt} \psi^{\varepsilon}(t)
=-\Delta\psi^{\varepsilon}(t) +
\frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)
\quad \quad \epsilon>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap
L^\infty(\mathbb{R}) \ . \] This equation represents a nonlinear Schr\"odinger
equation with a spatially concentrated nonlinearity. We show that in the limit
$\epsilon\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to
the weak dynamics of the NLS with point-concentrated nonlinearity: \[
i\frac{d}{dt} \psi(t) =H_{\alpha}\psi(t) . \] where $H_{\alpha}$ is the
laplacian with the nonlinear boundary condition at the origin
$\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)$ and
$\alpha=\int_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu\in
\mathbb{R}^+$ if $V \geq 0$ and for every $\mu\in (0,1)$ otherwise. The same
result holds true for a nonlinearity with an arbitrary number $N$ of
concentration points
