Nonlinear scattering conjecture for finite-energy Gross–Pitaevskii solutions

Let $\psi=1+u$ be a global solution of the Gross–Pitaevskii equation in $C(\mathbb{R};1+F_1)$, and let $E_1$ denote its renormalized energy. Fix the threshold $\mathcal{E}_0>0$ from the preceding definition, and let $M$ and $H$ be the transform and linear operator used in the scattering formulation. Then the nonlinear scattering conjecture. For every such solution satisfying

$$E_1(\psi)<\mathcal{E}_0,$$

there is a unique $z_+\in H^1(\mathbb{R}^3)$ such that

$$E_1(\psi)=\|\langle\nabla\rangle z_+\|_{L^2}^2$$

and

$$\|M(u(t))-e^{-itH}z_+\|_{H^1(\mathbb{R}^3)}\to0\qquad(t\to\infty).$$

Moreover, the map $u(0)\mapsto z_+$ is a homeomorphism between the open balls of radius $\mathcal{E}_0^{1/2}$ around $0$ in $F_1$ and $H^1$. This formulates the expected small-energy nonlinear scattering and asymptotic completeness statement; the supplied source does not provide evidence that it has been resolved, so its status remains open.