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 Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] 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.