We provide detailed analysis of the complex energy eigenvalue 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 quasi-bound states in continuum (or QBIC states). These are
resonant states with very long lifetimes 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 g6, where g is the coupling constant. As a result of
this expansion, we find that this state is a non-analytic effect resulting from
the van Hove singularity; it cannot be predicted from the ordinary perturbation
analysis that relies on Fermi's golden rule. We will also numerically
demonstrate the time evolution of the QBIC state using the effective potential
method in order to show the stability of the QBIC wave function in comparison
with that of the other eigenstates.Comment: Around 20 pages, 50 total figure