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

TL;DR

This paper demonstrates that in two and three dimensions, the nonlinear Schrödinger equation with a point interaction fails to scatter to free solutions or standing waves for low power nonlinearities, extending previous one-dimensional results.

Contribution

It extends the analysis of scattering failure for the NLS with delta potentials to higher dimensions, requiring new methods due to stronger singularities.

Findings

Failure of scattering in 2D and 3D for low power nonlinearities

Extension of 1D results to higher dimensions with new techniques

Different treatment of the linear part due to stronger singularities

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.