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A semiclassical method of complex trajectories for the calculation of thetunneling exponent in systems with many degrees of freedom is furtherdeveloped. It is supplemented with an easily implementable technique, whichenables one to single out the physically relevant trajectory from the whole setof complex classical trajectories. The method is applied to semiclassicaltransitions of a bound system through a potential barrier. We find that theproperties of physically relevant complex trajectories are qualitativelydifferent in the cases of potential tunneling at low energy and dynamicaltunneling at energies exceeding the barrier height. Namely, in the case of highenergies, the physically relevant complex trajectories describe tunneling viacreation of a state close to the top of the barrier. The method is checkedagainst exact solutions of the Schrodinger equation in a quantum mechanicalsystem of two degrees of freedom.Comment: 16 pages, 9 figures, version to appear in JET

Dynamical tunneling of bound systems through a potential barrier: Complex way to the top