| Abstract: |
| In this talk we consider the nonlinear Schr\odinger equation with a point nonlinearity in 1d, where the point nonlinearity is described
as a jump condition at a point in space in the linear Schr\odinger equation.
$H^1$ local well-posedness theory to this equation is available in [1], the authors in [2] established $L^2$ supercritical global existence, and blow-up
dichotomy. We address in this talk a $L^2$ supercritical scattering result applying the Kenig-Merle method [3],
but it is required to use an appropriate function space according to the smoothing properties of the corresponding Duhamel form,
which is in fact independent of the space variable.
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{\bf References.} \
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[1] R. Adami, A. Teta, \textit{A class of nonlinear Schr\odinger equations with
concentrated nonlinearity}, J. Funct. Anal. \textbf{180} (2001), no. 1, pp.
148-175. \
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[2] J. Holmer, C. Liu, \textit{Blow-up for the 1D nonlinear Schr\odinger
equation with point nonlinearity I: Basic theory}, arXiv:1510.03491 \
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[3] C. Kenig and F. Merle, \textit{Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schr\odinger
equation in the radial case,} Invent. Math. \textbf{166} (2006) no.3 645-675 \ |
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