Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS

Let $d\geq4$ and let u be a global solution to the focusing mass-critical nonlinear Schrodinger equation $iu_t+\Delta u=-|u|^{\frac{4}{d}}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state Q. We prove that if u does not scatter, then, up to phase rotation and scaling, u is the solitary wave $e^{it}Q$. Combining this result with that of Merle [Duke Math. J., 69 (1993), pp. 427–453], we obtain that in dimensions $d\geq4$, the only spherically symmetric minimal-mass nonscattering solutions are, up to phase rotation and scaling, the pseudoconformal ground state and the ground state solitary wave.