Resonant tunneling effect for quantum walks on directed graphs

Quantum walks that depend smoothly on a small parameter ε≥0 are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as ε→+0 is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for ε=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.