Traversal and tunneling time across barriers with jump discontinuities, and smooth barriers with compact support

Using the theory of time-of-arrival operators, we introduce the concept of \textit{partial} and \textit{full-tunneling} to determine the traversal and tunneling time of an incident wavepacket $\tilde{\psi}(k)$ across barriers with jump discontinuities, and smooth barriers with compact support. These two `times' are not necessarily equal. The \textit{full-tunneling} process occurs when the support of $\tilde{\psi}(k)$ is below the minimum height of the barrier, resulting to an instantaneous tunneling time. Meanwhile, the \textit{partial-tunneling} process occurs when the support or a segment of the support of $\tilde{\psi}(k)$ lies between the minimum and maximum height of the barrier. For this case, the particle does not ``fully'' tunnel through the entire barrier system resulting to a non-zero traversal time. We will show that potential barriers with jump discontinuities exhibit \textit{full tunneling} and \textit{partial tunneling}, while smooth barriers with compact support only exhibit \textit{partial tunneling}. The implications of these results on the resolution of the tunneling time problem are then discussed.