On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data

For the focusing mass-critical nonlinear Schrodinger equation $iu_t+Delta u=-|u|^{4/d}u$, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation $Delta Q-Q+Q^{1+frac 4d}=0$. Previous results in this direction were established in [12, 16, 17, 29] and they all require $u_0in H_x^1(mathbb{R}^d)$. In this paper, we consider the rigidity results for rough initial data $u_0 in H_x^s(mathbb{R}^d)$ for any $s>0$. We show that in dimensions $dge 4$ and under the radial assumption, the only solution that does not scatter in both time directions (including the finite time blowup case) must be global and coincide with the solitary wave $e^{it}Q$ up to symmetries of the equation. The proof relies on a non-uniform local iteration scheme, the refined estimate involving the $P^{pm}$ operator and a new smoothing estimate for spherically symmetric solutions.