Resonant tunneling effect for quantum walks on directed graphs

Quantum walks that depend smoothly on a small parameter $\varepsilon\ge0$ are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as $\varepsilon\to+0$ is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for $\varepsilon=0$ at points in the essential spectrum (the unit circle) that are asymptotically approached by a quantum resonance. Furthermore, a phenomenon resembling and extending the resonant tunneling effect is observed by analyzing this discrepancy through resonant states.