In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: $$i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) + \frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \quad \epsilon>0\ ,\quad V\in L^1(\erre,(1+|x|)dx) \cap L^\infty(\erre) \ .$$ 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 $H^1(\erre)$ 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 $\alpha=\int_{\erre}Vdx$. The convergence occurs for every $\mu\in \erre^+$ 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 %non superposing concentration points.

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

#### Abstract

In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: $$i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) + \frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \quad \epsilon>0\ ,\quad V\in L^1(\erre,(1+|x|)dx) \cap L^\infty(\erre) \ .$$ 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 $H^1(\erre)$ 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 $\alpha=\int_{\erre}Vdx$. The convergence occurs for every $\mu\in \erre^+$ 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 %non superposing concentration points.
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2014
nonlinear schroedinger
point interaction
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.14086/1527