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

Abstract

In this paper we consider the nonlinear Schrödinger (NLS) equation with power nonlinearity and a point interaction (a ‘δ-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 (2021 Discrete Contin. Dyn. Syst. 41 1507–17) consider the NLS equation with potentials and measures, singular enough to include the δ-potential in dimension one and they show analogous properties. We extend the result to higher dimensions and this needs a different treatment of the linear part of the interaction, due the qualitatively different and stronger character of the singularity involved.