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 $G^{\alpha }$ as a function of time, and we show that the error in adiabatic approximation remains small for running times of order $g^{-2}\mathrm{\mid }\ln g{\mathrm{\mid }}^{6\alpha }$. Here $g$ denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian.