  In quantum mechanics, systems can be described in phase space in terms of the
Wigner function and the star-product operation. Quantum characteristics, which
appear in the Heisenberg picture as the Weyl's symbols of operators of
canonical coordinates and momenta, can be used to solve the evolution equations
for symbols of other operators acting in the Hilbert space. To any fixed order
in the Planck's constant, many-body potential scattering problem simplifies to
a statistical-mechanical problem of computing an ensemble of quantum
characteristics and their derivatives with respect to the initial canonical
coordinates and momenta. The reduction to a system of ordinary differential
equations pertains rigorously at any fixed order in $\hbar$. We present
semiclassical expansion of quantum characteristics for many-body scattering
problem and provide tools for calculation of average values of time-dependent
physical observables and cross sections. The method of quantum characteristics
admits the consistent incorporation of specific quantum effects, such as
non-locality and coherence in propagation of particles, into the semiclassical
transport models. We formulate the principle of stationary action for quantum
Hamilton's equations and give quantum-mechanical extensions of the Liouville
theorem on the conservation of phase-space volume and the Poincar\'e theorem on
the conservation of $2p$ forms. The lowest order quantum corrections to the
Kepler periodic orbits are constructed. These corrections show the resonance
behavior.
