  A "quasiclassical" approximation to the quantum spectrum of the Schroedinger
equation is obtained from the trace of a quasiclassical evolution operator for
the "hydrodynamical" version of the theory, in which the dynamical evolution
takes place in the extended phase space $[q(t),p(t),M(t)] = [q_i, \partial_i S,
\partial_i \partial_j S ]$. The quasiclassical evolution operator is
multiplicative along the classical flow, the corresponding quasiclassical zeta
function is entire for nice hyperbolic flows, and its eigenvalue spectrum
contains the spectrum of the semiclassical zeta function. The advantage of the
quasiclassical zeta function is that it has a larger analyticity domain than
the original semiclassical zeta function; the disadvantage is that it contains
eigenvalues extraneous to the quantum problem. Numerical investigations
indicate that the presence of these extraneous eigenvalues renders the original
Gutzwiller-Voros semiclassical zeta function preferable in practice to the
quasiclassical zeta function presented here. The cumulant expansion of the
exact quantum mechanical scattering kernel and the cycle expansion of the
corresponding semiclassical zeta function part ways at a threshold given by the
topological entropy; beyond this threshold quantum mechanics cannot resolve
fine details of the classical chaotic dynamics.
