Yukawa model of screening in low-dimensional excitons: diagonalization, perturbation, variation, and resummation analysis

The Yukawa interaction describes Coulomb screening in several physical systems. We investigate bound quantum states of the Yukawa potential in low-dimensional structures as a model of e.g. excitons in semiconductor nanostructures. Diagonalization, perturbation, variation, and resummation methods are all applied to the problem and their accuracy is compared. For moderate positive screening, all methods are found to be applicable and, in particular, the variational approach is highly accurate except near the threshold for bound states. In contrast, only hypergeometric resummation captures the correct behaviour in the negative (antiscreened) regime. We also determine the dimensional dependence of the critical screening, above which bound states cease to exist. As an application, the critical screening is used to compute critical doping levels for the existence of bound excitons in quantum wells.