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On Schrodinger equation with point interactions

Raffaele Scandone
Friday, December 2, 2016 - 16:30

Singular perturbations of the Laplacian arise naturally in the context of quantum system of particles subject to interactions with certain fixed points. In three dimensions and with a single point interaction at the origin, there exists a one parameter family $-\Delta_{\alpha}$, $\alpha\in(-\infty,+\infty]$, of selfadjoint operators on $L^2$, which acts as the free Laplacian on functions that are supported away from the origin. In this talk, after introducing the definition and the main properties of these operators, we discuss two main topics:

  • the domains of the fractional powers of  $-\Delta_{\alpha}$, the analogues of the classical Sobolev space $H^s$  in the free case.
  • dispersive properties of the unitary group $e^{it\Delta_{\alpha}}$.

Eventually, we give some applications to the study of NLS equation with point interactions.

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