The charge transfer from an adatom to a semiconductor substrate of one-dimensional quantum dot array is evaluated theoretically. Due to the Van Hove singularity in the density of electron states at the band edges, the charge transfer decay rate is enhanced nonanalytically in terms of the coupling constant $g$ as $g^{4∕3}$. The optical absorption spectrum for the ionization of a core level electron of the adatom to the conduction band is also calculated. The reversible non-Markovian process and irreversible Markovian process in the time evolution of the adatom localized state manifest themselves in the absorption spectrum through the branch point and pole contributions, respectively.

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.

At the exceptional point where two eigenstates coalesce in open quantum systems, the usual diagonalization scheme breaks down and the Hamiltonian can only be reduced to the Jordan block form. Most of the studies on the exceptional point appearing in the literature introduce a phenomenological effective Hamiltonian that essentially reduces the problem to that of a finite non-Hermitian matrix for which it is straightforward to obtain the Jordan form. In this paper, we demonstrate how the microscopic total Hamiltonian of an open quantum system reduces to the Jordan block form at an exceptional point in an exact manner that treats the continuum without any approximation by extending the problem to include eigenstates with complex eigenvalues that reside outside the Hilbert space. Our method relies on the Brillouin-Wigner-Feshbach projection method according to which we can obtain a finite-dimensional effective Hamiltonian that shares the discrete sector of the spectrum with the total Hamiltonian. Because of the eigenvalue dependence of the effective Hamiltonian due to the dynamical nature of the coupling between the discrete states via the continuum states, a coalescence of eigenvalues results in the coalescence of the corresponding eigenvectors of the total Hamiltonian, which means that the system is at an exceptional point. We also introduce an extended Jordan form basis away from the exceptional point, which provides an alternative way to obtain the Jordan block at an exceptional point. The extended Jordan block connects continuously to the Jordan block exactly at the exceptional point implying that the observable quantities are continuous at the exceptional point.

We demonstrate the trapping of a conduction electron between two identical adatom impurities in a one-dimensional semiconductor quantum-dot array system (quantum wire). Bound steady states arise even when the energy of the adatom impurity is located in the continuous one-dimensional energy miniband. The steady state is a realization of the bound state in continuum (BIC) phenomenon first proposed by von Neuman and Wigner [Phys. Z. 30, 465 (1929)]. We analytically solve the dispersion equation for this localized state, which enables us to reveal the mechanism of the BIC. The appearance of the BIC state is attributed to the quantum interference between the impurities. The Van Hove singularity causes another type of bound state to form above and below the band edges, which may coexist with the BIC.

The decay rate $\gamma$ of an excited dipole molecule inside a waveguide is evaluated for the strongly coupled matter-field case near a cutoff frequency ${\omega }_c$ without using perturbation analysis. Because of the singularity in the density of photon states at the cutoff frequency, we find that $\gamma$ depends nonanalytically on the coupling constant $g$ as $g^{4/3}$, which leads to a vast increase in the decay rate.