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In this paper we consider the NLS equation with power nonlinearity and apoint interaction (a "$\delta$-potential" in the physical literature) indimension two and three. We will show that for low power nonlinearities thereis failure of scattering to the free dynamics or to standing waves. In therecent paper Murphy and Nakanishi consider the NLS equation with potentials andmeasures, singular enough to include the $\delta$-potential in dimension oneand they show analogous properties. In this respect our contribution is anextension to higher dimension and it needs a different treatment of the linearpart of the interaction, due the qualitatively different and stronger characterof the singularity involved.

Failure of scattering for the NLSE with a point interaction in dimension two and three