  We investigate theoretically population dynamics in an N-level system for
pulses of arbitrary shape. In order to use Floquet theory the pulse only has to
have finite support in time, which is well fulfilled in experiments.
Furthermore the dynamics must be approximately time-local, which restricts to
negligible memory effects. Despite the few requirements we are able to derive
criteria for resonances, which means that all population rests in a preselected
state after the pulse has been applied. We show that the necessary criterion on
the Floquet quasienergies is a generalization of the pi-pulse criterion,
derived within the Rotating Wave Approximation and adiabatic aproximation for
the envelope by Holthaus et. al., to pulses of arbitrary shape. Furthermore we
find that only the eigenvectors contain enough information to give a sufficient
criterion for a resonance. We constuct the resonant N-level propagator and show
that it is block diagonal, leading to a description by an effective
2-level-system composed of the initial and final state.
