A note on the switching adiabatic theorem

We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class G^α as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g^-2 |ln g |^6α. Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian.