  Phase space reflection operators lie at the core of the Wigner-Weyl
representation of density operators and observables. The role of the
corresponding classical reflections is known in the construction of
semiclassical approximations to Wigner functions of pure eigenstates and their
coarsegrained microcanonical superpositions, which are not restricted to
classically integrable systems. In their active role as unitary operators, they
generate transitions between pairs of eigenstates specified by transition
Wigner functions (or cross-Wigner functions): The square modulus of the
transition Wigner function at each point in phase space is the transition
probability for the reflection through that point.
  Coarsegraining the initial and final energies provides a transition
probability density as a phase space path integral. It is here investigated in
the simplest classical approximation involving microcanonical Wigner functions.
A reflection operator generates a transition between a pair of energy shells
with a probability density given by the integral of the inverse modulus of a
Poisson bracket over the intersection of a shell with the reflection of its
pair. The singularity of the pair of Wigner functions at their dominant
caustics is nicely integrable over their intersection, except for a single
degree of freedom. Even though this case is not directly relevant for future
investigations of chaotic systems, it is shown here how the improved
approximation of the spectral Wigner functions in terms of Airy functions
resolves the singularity.
