Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

TL;DR

This paper proves the scattering behavior of solutions to a focusing nonlinear Schrödinger equation with a repulsive Dirac delta potential at the critical mass-energy threshold, and shows the failure of uniform space-time bounds at this threshold.

Contribution

It establishes scattering results for the focusing NLS with a delta potential at the critical threshold and demonstrates the breakdown of uniform bounds there.

Findings

Solutions scatter at the mass-energy threshold.

Uniform space-time bounds fail at the threshold.

Results extend understanding of NLS with singular potentials.

Abstract

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schr\"odinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+\gamma\delta(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_{\gamma}(u_{0})[M(u_{0})]^{\sigma}=E_{0}(Q)[M(Q)]^{\sigma}$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_{\gamma}$ is the energy, $M$ is the mass and $\sigma=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy threshold.