  We consider the 1D nonlinear Schr\"odinger equation with focusing point
nonlinearity. "Point" means that the pure-power nonlinearity has an
inhomogeneous potential and the potential is the delta function supported at
the origin. This equation is used to model a Kerr-type medium with a narrow
strip in the optic fibre. There are several mathematical studies on this
equation and the local/global existence of solution, blow-up occurrence and
blow-up profile have been investigated. In this paper we focus on the
asymptotic behavior of the global solution, i.e, we show that the global
solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical
case. The main argument we use is due to Kenig-Merle, but it is required to
make use of an appropriate function space (not Strichartz space) according to
the smoothing properties of the associated integral equation.
