Tunneling time scale of under-the-barrier forerunners

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Time-dependent analytical solutions to Schrödinger's equation with quantum shutter initial conditions are used to investigate the issue of the tunneling time of forerunners in rectangular potential barriers. By using a time-frequency analysis, we find the existence of a regime characterized by the opacity of the barrier, where the maximum peak of a forerunner measured at the barrier transmission edge $x=L$ corresponds to a genuine tunneling process. The corresponding time scale represents the tunneling time of the forerunner through the classically forbidden region.
(1) Equation (1) was removed. (2) The first and second paragraphs in the second column of p.1 were modified. (3) In Fig. 1 we removed the plot of $t_p$. (4) We added an inset to Fig. 4

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