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We study stable blow-up dynamics in the $L^2$-critical nonlinearSchr\"{o}dinger equation in high dimensions. First, we show that in dimensions$d=4$ to $d=12$ generic blow-up behavior confirms the "log-log" regime in ournumerical simulations, including the log-log rate and the convergence of theblow-up profiles to the rescaled ground state; this matches the description ofthe stable blow-up regime in the dimension $d =2$ (for the 2d cubic NLSequation).  Next, we address the question of rigorous justification of the "log-log"dynamics in higher dimensions ($d \geq5)$, at least for the initial data withthe mass slightly larger than the mass of the ground state, for which thespectral conjecture has yet to be proved, see [34] and [10]. We give anumerically-assisted proof of the spectral property for the dimensions from$d=5$ to $d=12$, and a modification of it in dimensions $2 \leq d \leq 12$.This, combined with previous results of Merle-Rapha\"el, proves the "log-log"stable blow-up regime in dimensions $d \leq 10$ and radially stable for $d \leq12$.

Blow-up dynamics and spectral property in the $L^2$-critical nonlinear Schr\"odinger equation in high dimensions