  The mixed density operator for coarsegrained eigenlevels of a static
Hamiltonian is represented in phase space by the spectral Wigner function,
which has its peak on the corresponding classical energy shell. The action of
trajectory segments along the shell determine the phase of the Wigner
oscillations in its interior. The classical transitions between any pair of
energy shells, driven by a general external time dependent Hamiltonian, also
have a smooth probability density. It is shown here that a further contribution
to the transition between the corresponding pair of coarsegrained energy
levels, which oscillates with either energy, or the driving time, is determined
by four trajectory segments (two in the pair of energy shells and two generated
by the driving Hamiltonian) that join exactly to form a closed compound orbit.
In its turn, this sequence of segments belongs to the semiclassical expression
of a compound unitary operator that combines four quantum evolutions: a pair
generated by the static internal Hamiltonian and a pair generated by the
driving Hamiltonian. The closed compound orbits are shown to belong to
continuous families, which are initially seeded at points where the classical
flow generated by both Hamiltonians commute.
