  The phase space Koopman-van Hove (KvH) equation can be derived from the
asymptotic semiclassical analysis of partial differential equations.
Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase
factor and the transport equation for the amplitude. These two equations can be
combined to form a nonlinear semiclassical version of the KvH equation in
configuration space. There is a natural injection of configuration space
solutions into phase space and a natural projection of phase space solutions
onto configuration space. Hence, every solution of the configuration space KvH
equation satisfies both the semiclassical phase space KvH equation and the
Hamilton-Jacobi constraint. For configuration space solutions, this constraint
resolves the paradox that there are two different conserved densities in phase
space. For integrable systems, the KvH spectrum is the Cartesian product of a
classical and a semiclassical spectrum. If the classical spectrum is
eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin
(JWKB) matching conditions, the semiclassical spectrum satisfies the
Einstein-Brillouin-Keller quantization conditions which include the correction
due to the Maslov index. However, semiclassical analysis uses different choices
for boundary conditions, continuity requirements, and the domain of definition.
For example, use of the complex JWKB method allows for the treatment of
tunneling through the complexification of phase space. Finally, although KvH
wavefunctions include the possibility of interference effects, interference is
not observable when all observables are approximated as local operators on
phase space. Observing interference effects requires consideration of nonlocal
operations, e.g. through higher orders in the asymptotic theory.
