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

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.