In this dissertation we will study a wide range of phenomena from atomic, molecular, and optical to solid-state physics. We will find a common theme in problems from these different branches of physics in that they can all be modeled by some variation of a simple bi-linear Hamiltonian. Each of these models will also share a key feature in that they all contain one or more singularities (called a Van Hove singularity in the context of solid-state) in the density of allowed states associated with a branch point that results near the edge of a continuous energy spectrum. In addition, the fact that each of these models is one-dimensional will maximize the effect of the singularity on the system. We will show that when a discrete state is coupled with the continuum that in the vicinity of the singularity Fermi's golden rule breaks down; the golden rule normally predicts that the de-excitation rate of the discrete state should be proportional to g² where gis the dimensionless coupling constant between the discrete state and the continuum. Relying on a non-perturbative approach, we will show that the de-excitation rate is actually proportional to g^4/3 in the vicinity of the singularity. This results in a dramatic amplification of the decay rate.

In the main topic of the dissertation, we will consider a nano-scale semiconductor superlattice with either a single impurity site or multiple impurities (which behave as electron donors or acceptors) in which there are two Van Hove singularities in the density of electron states which occur at the two edges of the conduction band. These singularities result in the non-analytic g^4/3 amplification of the charge transfer rate from the discrete impurity site into the electronic conduction band where gis the coupling constant between the impurity state and the conduction band.

We will demonstrate other results including an asymmetry in the optical absorption profile for monochromatic light incident on a core electron state in the single impurity system and bound states in continuum (BIC) for the superlattice system with two impurities.