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

Abstract

In this paper we consider the NLS equation with power nonlinearity and a point interaction (a "$\delta$-potential" in the physical literature) in dimension two and three. We will show that for low power nonlinearities there is failure of scattering to the free dynamics or to standing waves. In the recent paper Murphy and Nakanishi consider the NLS equation with potentials and measures, singular enough to include the $\delta$-potential in dimension one and they show analogous properties. In this respect our contribution is an extension to higher dimension and it needs a different treatment of the linear part of the interaction, due the qualitatively different and stronger character of the singularity involved.