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

In the present paper we study the following scaled nonlinear Schr"odinger equation (NLS) in one space dimension: $$i\frac{d}{dt}{\psi }^{\epsilon }(t)=-\mathrm{\Delta }{\psi }^{\epsilon }(t)+\frac{1}{ϵ}V\left(\frac{x}{ϵ}\right)\mathrm{\mid }{\psi }^{\epsilon }(t){\mathrm{\mid }}^{2\mu }{\psi }^{\epsilon }(t)ϵ>0,V\in L^1(\mathbb{R},(1+\mathrm{\mid }x\mathrm{\mid })dx)\cap L^{\mathrm{\infty }}(\mathbb{R})\mathrm{.}$$ This equation represents a nonlinear Schr"odinger 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: $$i\frac{d}{dt}\psi (t)=H_{\alpha }\psi (t)\mathrm{.}$$ where $H_{\alpha }$ is the laplacian with the nonlinear boundary condition at the origin ${\psi }^{\mathrm{\prime }}(t,0+)-{\psi }^{\mathrm{\prime }}(t,0-)=\alpha \mathrm{\mid }\psi (t,0){\mathrm{\mid }}^{2\mu }\psi (t,0)$ and $\alpha ={\int }_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu\in $\mathbb{R}$^+$if$V \geq 0$andforevery$\mu\in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration points