The semiclassical approximation for the energy transition probability density
in a recent sequence of papers depends on the full unitary operator that has
driven the transition and on the trajectories of the driven classical
Hamiltonian. Neither of these is explicitly given for a transition generated by
a general Hamiltonian, even if it is time-independent, so that a sudden
transition was presumed. The theory is here generalized for arbitrary driving
Hamiltonians, by basing it on a compound unitary operator that combines four
evolutions, a pair generated by the original Hamiltonian and a pair generated
by the driving Hamiltonian. The supporting classical structure is again that of
closed compound orbits, but now these are composed of four trajectory segments,
corresponding to the quantum evolutions. Notwithstanding the increased
complexity, all underlying trajectory segments are then generated by
Hamiltonians that are known a priory.
The phase space integral for the smooth classical background, with respect to
variations of the pair of energies, is preserved from the previous papers. The
quantum oscillations of the transition density again result from the stationary
phase approximation of a double Fourier integral over the trace of the
semiclassical compound unitary operator. Even though it now combines four
evolutions instead of two, the phases of the oscillations agree with the
previous results if the driven Hamiltonian is known