Two-channel quantum wire with an adatom impurity: Role of the van Hove singularity in the quasibound state in continuum, decay rate amplification, and the Fano effect

We provide detailed analysis of the complex eigenenergy spectrum for a two-channel quantum wire with an attached adatom impurity. The study is based on our previous work [Phys. Rev. Lett. 99, 210404 (2007)], in which we presented the quasibound states in continuum (or QBIC states). These are resonant states with very long lifetime that form as a result of two overlapping continuous energy bands, one of which, at least, has a divergent van Hove singularity at the band edge. We provide analysis of the full energy spectrum for all solutions, including the QBIC states, and obtain an expansion for the complex eigenvalue of the QBIC state. We show that it has a small decay rate of the order g(6), where g is the coupling constant of the adatom impurity. As a result of this expansion, we find that this state is a nonanalytic effect resulting from the van Hove singularity; it cannot be predicted from the ordinary perturbation analysis that relies on Fermi's golden rule. We also show that the QBIC state emerges as a direct result of the destabilization of the stable state that often exists on the outside edge of a band due to the divergence. As another result of the van Hove singularity, it has been previously reported that the decay rate of an unstable state is amplified in the vicinity of the band edge such that it is proportional to g(4/3). This again results from a breakdown of the Fermi rule. Here we explicitly show how the system behaves in the crossover region between the g(4/3) region and the Fermi region. Finally, we calculate the local density of states near the adatom. We are able to demonstrate that the interference between two unstable states with a very large decay rate and one unstable state with a small decay rate results in a characteristic asymmetric Fano profile. This effect leads to the best chance of detecting the QBIC by scanning tunneling microscopy probe.