Excitons in quantum wires

This paper deals with excitons in quantum wires. We first study these excitons as the limit of excitons in D dimensions when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D \rightarrow 1$\end{document}. In order to do it, we have had to find a new resolution of the hydrogen atom Schrödinger equation: besides the fact that the usual resolution found in textbooks is not valid for D exactly equal to 1, it is, surprisingly enough, inconsistent since it relies on two hypergeometric functions which are not independent for the parameters of physical interest! In a second part, we write down the exact potential felt by the exciton relative motion along the wire in terms of the wire confinement. This allows a quite precise determination of the effective Coulomb potential for this 1D motion, which is of crucial importance to obtain a meaningfull finite value for the exciton ground state energy. In a last part, we study the dependence of the exciton energies on the wire area and anisotropy. While the quantitative results are here given for cylindrica l and rectangular wires with infinite barriers, we show how they can easily be extended to any particular wire shape and barrier height.