Upper bounds in non-autonomous quantum dynamics

We prove upper bounds on outside probabilities for generic non-autonomous Schr\"odinger operators on lattices of arbitrary dimension. Our approach is based on a combination of commutator method originated in scattering theory and novel monotonicity estimate for certain mollified asymptotic observables that track the spacetime localization of evolving states. Sub-ballistic upper bounds are obtained, assuming that momentum vanishes sufficiently fast in the front of the wavepackets. A special case gives a refinement of the general ballistic upper bound of Radin-Simon's, showing that the evolution of wavepackets are effectively confined to a strictly linear light cone with explicitly bounded slope. All results apply to long-range Hamiltonian with polynomial decaying off-diagonal terms and can be extended, via a frozen-coefficient argument, to generic nonlinear Schr\"odinger equations on lattices.