##### Abstract

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(alpha) as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g(-2) vertical bar ln g vertical bar(6 alpha). Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4748968]