  A semiclassical method of complex trajectories for the calculation of the
tunneling exponent in systems with many degrees of freedom is further
developed. It is supplemented with an easily implementable technique, which
enables one to single out the physically relevant trajectory from the whole set
of complex classical trajectories. The method is applied to semiclassical
transitions of a bound system through a potential barrier. We find that the
properties of physically relevant complex trajectories are qualitatively
different in the cases of potential tunneling at low energy and dynamical
tunneling at energies exceeding the barrier height. Namely, in the case of high
energies, the physically relevant complex trajectories describe tunneling via
creation of a state close to the top of the barrier. The method is checked
against exact solutions of the Schrodinger equation in a quantum mechanical
system of two degrees of freedom.
