Time-ordered exponential ($U(t,t_0)=\mathcal{T}\exp [-i{\int }_{t_0}^{t}H(s)ds/\hslash ]$)

The time-ordered exponential solves $i\hslash {\partial }_{t}U=H(t)U$ with initial condition $U(t_0,t_0)=I$. Its expansion is the Dyson series. For bounded continuous Hamiltonian operators the series converges, as in differentiation of the Dyson time-ordered exponential. If the generators commute at all times, time ordering is unnecessary and an ordinary operator exponential suffices.