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The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

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