The equivalence between Floquet formalism and the multi-step approach in computing the evolution operator of a periodical time-dependent Hamiltonian

A theoretical proof is given to show that, under certain conditions which are easily satisfied in practice, the Floquet formalism when used to calculate spectral lineshapes is numerically equivalent to the multi-step approach in calculating the time evolution of a periodical time-dependent Hamiltonian. It is found that in Floquet formalism the perturbation expansion upto the N/2-th order gives the same result as the multi-step method which divides the period into N equal intervals.