Letters in Mathematical Physics

, Volume 104, Issue 12, pp 1557–1570

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

• Claudio Cacciapuoti
• Domenico Finco
• Diego Noja
• Alessandro Teta
Article

Abstract

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:
$$i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$
This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit $${\varepsilon\to 0}$$ the weak (integral) dynamics converges in $${H^1(\mathbb{R})}$$ to the weak dynamics of the NLS with point-concentrated nonlinearity:
$$i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$
where H α 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 $${\alpha=\int_{\mathbb{R}}V{\rm d}x}$$. The convergence occurs for every $${\mu\in \mathbb{R}^+}$$ if V ≥  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.

Keywords

Nonlinear Schrödinger equation nonlinear delta interactions zero-range limit of concentrated nonlinearities

Mathematics Subject Classification

81Q15 35B25 35B35 35B55

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Authors and Affiliations

• Claudio Cacciapuoti
• 1
• 5
• Domenico Finco
• 2
• Diego Noja
• 3
• Alessandro Teta
• 4
1. 1.Hausdorff Center for Mathematics, Institut für Angewandte MathematikBonnGermany
2. 2.Facoltà di IngegneriaUniversità Telematica Internazionale UninettunoRomeItaly
3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanItaly
4. 4.Dipartimento di Matematica G. CastelnuovoSapienza Università di RomaRomeItaly
5. 5.Dipartimento di Scienza e Alta TecnologiaUniversità dell’InsubriaComoItaly