  A comparison of classical and quantum evolution usually involves a
quasi-probability distribution as a quantum analogue of the classical phase
space distribution. In an alternate approach that we adopt here, the classical
density is projected on to the configuration space. We show that for billiards,
the eigenfunctions of the coarse-grained projected classical evolution operator
are identical to a first approximation to the quantum Neumann eigenfunctions.
However, even though there exists a correspondence between the respective
eigenvalues, their time evolutions differ. This is demonstrated numerically for
the stadium and lemon shaped billiards.
