article published yes Rowan Killip author Dong Li author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca Xiaoyi Zhang author Let 𝑑 ≥4 and let u be a global solution to the focusing mass-critical nonlinear Schrödinger equation 𝑖⁢𝑢𝑡 +Δ⁢𝑢 =−|𝑢|4𝑑 ⁢𝑢 with spherically symmetric 𝐻1𝑥 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 𝑒𝑖⁢𝑡⁢𝑄. Combining this result with that of Merle [Duke Math. J., 69 (1993), pp. 427–453], we obtain that in dimensions 𝑑 ≥4, 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. Society for Industrial & Applied Mathematics2009 eng 0036-1410 1095-7154 0804.112410.1137/080720358 411219-236 yes Killip, Rowan, Dong Li, Monica Vişan, and Xiaoyi Zhang. “Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics, 2009. <a href="https://doi.org/10.1137/080720358">https://doi.org/10.1137/080720358</a>. Killip, R., Li, D., Vişan, M., &#38; Zhang, X. (2009). Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/080720358">https://doi.org/10.1137/080720358</a> R. Killip, D. Li, M. Vişan, and X. Zhang, “Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 41, no. 1. Society for Industrial &#38; Applied Mathematics, pp. 219–236, 2009. Killip, Rowan, et al. “Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 41, no. 1, Society for Industrial &#38; Applied Mathematics, 2009, pp. 219–36, doi:<a href="https://doi.org/10.1137/080720358">10.1137/080720358</a>. Killip R, Li D, Vişan M, Zhang X. Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS. <i>SIAM Journal on Mathematical Analysis</i>. 2009;41(1):219-236. doi:<a href="https://doi.org/10.1137/080720358">10.1137/080720358</a> R. Killip, D. Li, M. Vişan, X. Zhang, SIAM Journal on Mathematical Analysis 41 (2009) 219–236. Killip R, Li D, Vişan M, Zhang X. 2009. Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS. SIAM Journal on Mathematical Analysis. 41(1), 219–236. 220782026-06-19T08:25:50Z2026-06-30T11:48:07Z