Letters in Mathematical Physics

, Volume 104, Issue 12, pp 1557–1570 | Cite as

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 Hα 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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Copyright information

© Springer Science+Business Media Dordrecht 2014

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

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